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Hemimean family: Difference between revisions

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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.955{{c}}, ~28/25 = 193.650{{c}}
* [[WE]]: ~2 = 1199.8194{{c}}, ~3/2 = 702.1353{{c}}, ~28/25 = 193.7425{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.112{{c}}, ~28/25 = 193.717{{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]]:  
Line 52: Line 54:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.720{{c}}, ~28/25 = 193.554{{c}}
* [[WE]]: ~2 = 1200.0098{{c}}, ~3/2 = 701.7170{{c}}, ~28/25 = 193.5520{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.714{{c}}, ~28/25 = 193.552{{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]]:  
Line 76: Line 80:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 701.822{{c}}, ~28/25 = 193.582{{c}}
* WE: ~2 = 1199.9154{{c}}, ~3/2 = 701.7875{{c}}, ~28/25 = 193.5853{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 701.835{{c}}, ~28/25 = 193.596{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8346{{c}}, ~28/25 = 193.5962{{c}}


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


Badness (Sintel): 1.034
Badness (Sintel): 1.03


=== Bellowblog ===
=== Bellowblog ===
Line 91: Line 95:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.567{{c}}, ~28/25 = 193.249{{c}}
* WE: ~2 = 1200.000{{c}}, ~3/2 = 702.5857{{c}}, ~28/25 = 193.2930{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.634{{c}}, ~28/25 = 193.293{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.6342{{c}}, ~28/25 = 193.2932{{c}}


{{Optimal ET sequence|legend=0| 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d }}
{{Optimal ET sequence|legend=0| 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d }}


Badness (Sintel): 1.183
Badness (Sintel): 1.18


== Siebog ==
== Siebog ==
Line 107: Line 111:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.164{{c}}, ~28/25 = 193.865{{c}}
* [[WE]]: ~2 = 1199.5790{{c}}, ~3/2 = 701.8397{{c}}, ~28/25 = 194.0111{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.723{{c}}, ~28/25 = 193.995{{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]]:  
Line 117: Line 123:
{{Optimal ET sequence|legend=1| 12e, 18e, 19, 31, 68e, 80, 99e, 130, 210e, 241, 340ce, 371ce, 470cdee, 501cde, 581cdee, 711ccdee }}
{{Optimal ET sequence|legend=1| 12e, 18e, 19, 31, 68e, 80, 99e, 130, 210e, 241, 340ce, 371ce, 470cdee, 501cde, 581cdee, 711ccdee }}


[[Badness]] (Sintel): 1.045
[[Badness]] (Sintel): 1.05


== Triglav ==
== Triglav ==
Line 124: Line 130:
[[Comma list]]: 3025/3024, 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|legend=1| 1 0 -2 -8 0 | 0 1 2 5 2 | 0 0 4 10 1 }}
: Mapping generators: ~2, ~3, ~18/11
: Mapping generators: ~2, ~3, ~11/9


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.288{{c}}, ~18/11 = 854.313{{c}}
* [[WE]]: ~2 = 1199.8764{{c}}, ~3/2 = 702.4302{{c}}, ~11/9 = 345.5856{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.407{{c}}, ~18/11 = 854.350{{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 }}
{{Optimal ET sequence|legend=1| 24d, 31, 80, 87, 111, 118, 198, 316, 514c, 545c }}
Line 140: Line 148:
Comma list: 352/351, 1001/1000, 3025/3024
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: {{Mapping| 1 0 -2 -8 0 5 | 0 1 2 5 2 -1 | 0 0 4 10 1 1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.707{{c}}, ~18/11 = 854.537{{c}}
* WE: ~2 = 1199.6554{{c}}, ~3/2 = 702.8049{{c}}, ~11/9 = 345.3412{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.937{{c}}, ~18/11 = 854.554{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9366{{c}}, ~11/9 = 345.4458{{c}}


{{Optimal ET sequence|legend=0| 24d, 31, 80, 87, 111, 118, 198 }}
{{Optimal ET sequence|legend=0| 24d, 31, 56, 80, 87, 111, 118, 167, 198 }}


Badness (Sintel): 1.159
Badness (Sintel): 1.16


== Semihemimean ==
== Semihemimean ==
Named by [[Xenllium]] in 2026, semihemimean tempers out [[9801/9800]] and splits the octave in halves.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~99/70 = 600.000{{c}}, ~3/2 = 702.002{{c}}, ~28/25 = 193.633{{c}}
* [[WE]]: ~99/70 = 599.9102{{c}}, ~3/2 = 702.1314{{c}}, ~28/25 = 193.7431{{c}}
* [[CWE]]: ~99/70 = 600.000{{c}}, ~3/2 = 702.135{{c}}, ~28/25 = 193.712{{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, 50, 68, 80, 118, 130, 198 }}
{{Optimal ET sequence|legend=1| 12, …, 38d, 50, 68, 80, 118, 130, 198, 248, 328, 446, 774c }}


[[Badness]] (Sintel): 1.787
[[Badness]] (Sintel): 1.79


=== 13-limit ===
=== 13-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000{{c}}, ~3/2 = 701.838{{c}}, ~28/25 = 193.671{{c}}
* WE: ~99/70 = 599.8791{{c}}, ~3/2 = 702.1476{{c}}, ~28/25 = 193.7877{{c}}
* CWE: ~99/70 = 600.000{{c}}, ~3/2 = 702.174{{c}}, ~28/25 = 193.787{{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 }}
{{Optimal ET sequence|legend=0| 12, …, 50, 68, 80, 118, 130, 198, 328, 576cf, 774cf, 904cef }}


[[Badness]] (Sintel): 1.550
Badness (Sintel): 1.55


=== 17-limit ===
=== 17-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~3/2 = 702.108{{c}}, ~28/25 = 193.723{{c}}
* WE: ~17/12 = 599.9481{{c}}, ~3/2 = 702.2594{{c}}, ~28/25 = 193.7763{{c}}
* CWE: ~17/12 = 600.000{{c}}, ~3/2 = 702.269{{c}}, ~28/25 = 193.776{{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 }}
{{Optimal ET sequence|legend=0| 12, …, 50, 68, 80, 118, 130, 198 }}


[[Badness]] (Sintel): 1.743
Badness (Sintel): 1.74


=== 19-limit ===
=== 19-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~3/2 = 702.252{{c}}, ~19/17 = 193.758{{c}}
* WE: ~17/12 = 599.9687{{c}}, ~3/2 = 702.3501{{c}}, ~19/17 = 193.7922{{c}}
* CWE: ~17/12 = 600.000{{c}}, ~3/2 = 702.355{{c}}, ~19/17 = 193.792{{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 }}
{{Optimal ET sequence|legend=0| 12, …, 50, 68, 80, 118, 130, 198 }}


[[Badness]] (Sintel): 1.318
Badness (Sintel): 1.32


== Subgroup extensions ==
== Subgroup extensions ==
=== Hemimean orion (2.3.5.7.17) ===
=== 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.  
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]])}.
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
Subgroup: 2.3.5.7.17
Line 224: Line 236:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.196{{c}}, ~28/25 = 193.655{{c}}
* WE: ~2 = 1199.7919{{c}}, ~3/2 = 702.2561{{c}}, ~28/25 = 193.7586{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.304{{c}}, ~28/25 = 193.737{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3042{{c}}, ~28/25 = 193.7365{{c}}


{{Optimal ET sequence|legend=1| 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328h, 446, 545c, 873cg }}
{{Optimal ET sequence|legend=0| 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328, 446, 545c, 873cg }}


Badness (Sintel): 0.884
Badness (Sintel): 0.884
Line 239: Line 251:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.132{{c}}, ~19/17 = 193.647{{c}}
* WE: ~2 = 1199.8239{{c}}, ~3/2 = 702.1623{{c}}, ~19/17 = 193.7325{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.213{{c}}, ~19/17 = 193.716{{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 }}
{{Optimal ET sequence|legend=0| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}
Line 247: Line 259:


=== Semiorion (2.3.5.7.17) ===
=== 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]])}.  
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
Subgroup: 2.3.5.7.17
Line 257: Line 269:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~3/2 = 702.347{{c}}, ~28/25 = 193.650{{c}}
* WE: ~17/12 = 600.0551{{c}}, ~3/2 = 702.1998{{c}}, ~28/25 = 193.5929{{c}}
* CWE: ~17/12 = 600.000{{c}}, ~3/2 = 702.218{{c}}, ~28/25 = 193.604{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 702.2183{{c}}, ~28/25 = 193.6044{{c}}


{{Optimal ET sequence|legend=1| 12, 30d, 38d, 50, 62, 68, 106d, 118, 248g, 316g }}
{{Optimal ET sequence|legend=0| 12, 30d, 38d, 50, 62, 68, 106d, 118, 248g, 316g }}


Badness (Sintel): 1.690
Badness (Sintel): 1.69


==== 2.3.5.7.17.19 subgroup ====
==== 2.3.5.7.17.19 subgroup ====
Line 272: Line 284:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~3/2 = 702.509{{c}}, ~19/17 = 193.669{{c}}
* WE: ~17/12 = 600.0873{{c}}, ~3/2 = 702.2450{{c}}, ~19/17 = 193.5751{{c}}
* CWE: ~17/12 = 600.000{{c}}, ~3/2 = 702.279{{c}}, ~19/17 = 193.592{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 702.2791{{c}}, ~19/17 = 193.5923{{c}}


{{Optimal ET sequence|legend=0| 12, …, 50, 68, 106d, 118, 248g, 316g }}
{{Optimal ET sequence|legend=0| 12, 30dh, 38d, 50, 68, 106d, 118, 248g, 316g }}


Badness (Sintel): 0.722
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:Rank 3]]
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