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Hemimean family: Difference between revisions

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The '''hemimean family''' of temperaments are rank-3 temperaments tempering out [[3136/3125]].  
{{Technical data page}}
The '''hemimean family''' of [[temperament]]s are [[rank-3 temperament]]s which [[temper out]] [[3136/3125]].  


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


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


[[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]]: [{{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]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9550, ~28/25 = 193.6499


[[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]]: 0.160 × 10<sup>-3</sup>
Line 32: Line 37:


=== Hemimean orion ===
=== Hemimean orion ===
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.  
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
Subgroup: 2.3.5.7.17
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Comma list: 1701/1700, 3136/3125
Comma list: 1701/1700, 3136/3125


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


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1960, ~28/25 = 193.6548
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1960, ~28/25 = 193.6548


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


Badness: 0.573
Badness: 0.573
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Comma list: 476/475, 1216/1215, 1445/1444
Comma list: 476/475, 1216/1215, 1445/1444


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


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~19/17 = 193.647
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~19/17 = 193.647


Optimal GPV sequence: {{Val list| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}
{{Optimal ET sequence|legend=1| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}


Badness: 0.456
Badness: 0.456


=== Semiorion ===
=== Semiorion ===
Semiorion is an alternative subgroup extension of lower complexity, which splits the octave into two.  
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
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Comma list: 289/288, 3136/3125
Comma list: 289/288, 3136/3125


Sval mapping: [{{val| 2 0 0 -6 5 }}, {{val| 0 1 0 0 1 }}, {{val| 0 0 2 5 0 }}]
Sval mapping: {{mapping| 2 0 0 -6 5 | 0 1 0 0 1 | 0 0 2 5 0 }}


Sval mapping generators: ~17/12, ~3, ~56/25
: sval mapping generators: ~17/12, ~3, ~56/25


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


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


Badness: 1.095
Badness: 1.095
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Comma list: 289/288, 361/360, 476/475
Comma list: 289/288, 361/360, 476/475


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


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


Optimal GPV sequence: {{Val list| 12, …, 50, 68, 106d, 118, 248g, 316g }}
{{Optimal ET sequence|legend=1| 12, …, 50, 68, 106d, 118, 248g, 316g }}


Badness: 0.569
Badness: 0.569
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[[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: [{{val| 0 -2 2 5 4 }}, {{val| 0 -1 0 0 -2 }}]
Mapping to lattice: {{mapping| 0 -2 2 5 4 | 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]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7205, ~28/25 = 193.5545


[[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 basis]]: 2.9/7.11/5
: [[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]]: 0.609 × 10<sup>-3</sup>
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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 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.8219, ~28/25 = 193.5816


Optimal GPV sequence: {{Val list| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}
{{Optimal ET sequence|legend=1| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}


Badness: 1.11 × 10<sup>-3</sup>
Badness: 1.11 × 10<sup>-3</sup>
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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 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.5667, ~28/25 = 193.2493


Optimal GPV sequence: {{Val list| 12f, 19e, 31, 56, 68e, 87, 118, 205d, 263f, 304f, 391df, 509df }}
{{Optimal ET sequence|legend=1| 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d }}


Badness: 1.26 × 10<sup>-3</sup>
Badness: 1.26 × 10<sup>-3</sup>
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[[Comma list]]: 540/539, 3136/3125
[[Comma list]]: 540/539, 3136/3125


[[Mapping]]: [{{val| 1 0 0 -3 8 }}, {{val| 0 1 0 0 3 }}, {{val| 0 0 2 5 -8 }}]
{{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 generators: ~2, ~3, ~768/343
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.1636, ~28/25 = 193.8645


[[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 basis]]: 2.3.11/10
: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.11/5


{{Val list|legend=1| 12e, 19, 31, 68e, 80, 99e, 130, 241, 340ce, 371ce, 470cde, 711cde }}
{{Optimal ET sequence|legend=1| 12e, 18e, 19, 31, 68e, 80, 99e, 130, 210e, 241, 340ce, 371ce, 470cdee, 501cde, 581cdee, 711ccdee }}


[[Badness]]: 0.870 × 10<sup>-3</sup>
[[Badness]]: 0.870 × 10<sup>-3</sup>
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[[Comma list]]: 3025/3024, 3136/3125
[[Comma list]]: 3025/3024, 3136/3125


[[Mapping]]: [{{val| 1 0 2 2 1 }}, {{val| 0 1 2 5 2 }}, {{val| 0 0 -4 -10 -1 }}]
{{Mapping|legend=1| 1 0 2 2 1 | 0 1 2 5 2 | 0 0 -4 -10 -1 }}
 
: mapping generators: ~2, ~3, ~18/11


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


{{Val list|legend=1| 31, 80, 87, 111, 118, 198, 316, 545c, 861ce }}
{{Optimal ET sequence|legend=1| 24d, 31, 80, 87, 111, 118, 198, 316, 514c, 545c }}


[[Badness]]: 0.819 × 10<sup>-3</sup>
[[Badness]]: 0.819 × 10<sup>-3</sup>


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Hemimean family| ]] <!-- main article -->
[[Category:Hemimean family| ]] <!-- main article -->
[[Category:Hemimean]]
[[Category:Hemimean]]
[[Category:Rank 3]]
[[Category:Rank 3]]
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