Hemimean family: Difference between revisions

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@@ Line 36: / Line 36: @@
 [[Projection pair]]s: <code>5 3136/625, 7 68841472/9765625</code> to 2.3.25/7
-=== 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
-Subgroup-val mapping: {{mapping| 1 0 0 -3 -5 | 0 1 0 0 5 | 0 0 2 5 1 }}
-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 ET sequence|legend=1| 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: {{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{{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 ET sequence|legend=0| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}
-Badness (Sintel): 0.578
-=== 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
-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:
-* CTE: ~17/12 = 600.000{{c}}, ~3/2 = 702.347{{c}}, ~28/25 = 193.650{{c}}
-* CWE: ~17/12 = 600.000{{c}}, ~3/2 = 702.218{{c}}, ~28/25 = 193.604{{c}}
-{{Optimal ET sequence|legend=1| 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: {{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{{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}}
-{{Optimal ET sequence|legend=0| 12, …, 50, 68, 106d, 118, 248g, 316g }}
-Badness (Sintel): 0.722
 == Belobog ==
@@ Line 277: / Line 210: @@
 [[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 }}
+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 ET sequence|legend=1| 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: {{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{{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 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:
+* CTE: ~17/12 = 600.000{{c}}, ~3/2 = 702.347{{c}}, ~28/25 = 193.650{{c}}
+* CWE: ~17/12 = 600.000{{c}}, ~3/2 = 702.218{{c}}, ~28/25 = 193.604{{c}}
+{{Optimal ET sequence|legend=1| 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: {{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{{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}}
+{{Optimal ET sequence|legend=0| 12, …, 50, 68, 106d, 118, 248g, 316g }}
+Badness (Sintel): 0.722
 [[Category:Temperament families]]
 [[Category:Hemimean family| ]] <!-- main article -->
 [[Category:Rank 3]]