Hemimean family: Difference between revisions

The '''hemimean family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] 3136/3125, the [[hemimean comma]].  

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]])².

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125

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

[[Mapping to lattice]]: {{mapping| 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

: [[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]]:

* [[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 }}]

: [[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 }}

[[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 pair]]s: <code>5 3136/625, 7 68841472/9765625</code> to 2.3.25/7

[[Subgroup]]: 2.3.5.7.11

[[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 }}

: Angle (28/25, 16/15) = 93.2696

: [[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 }}

: [{{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, 19e, 31, 68e, 87, 99e, 118, 130, 217, 248 }}

[[Badness]] (Sintel): 0.732

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

Scales: [[belobog31]]

Subgroup: 2.3.5.7.11.13

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

Mapping: {{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{{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}}

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

Badness (Sintel): 1.03

Subgroup: 2.3.5.7.11.13

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

Mapping: {{mapping| 1 0 0 -3 -9 -4 | 0 1 0 0 2 -1 | 0 0 2 5 8 8 }}

* 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}}

Badness (Sintel): 1.18

[[Subgroup]]: 2.3.5.7.11

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

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

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

[[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}}

[[Minimax tuning]]:  

: [{{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| 12e, 18e, 19, 31, 68e, 80, 99e, 130, 210e, 241, 340ce, 371ce, 470cdee, 501cde, 581cdee, 711ccdee }}

[[Badness]] (Sintel): 1.05

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 3025/3024, 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 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.

 

 

 

 

 

 

 

 

 

 

 

 

 

Badness (Sintel): 0.578

 

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]])}.

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Category:Hemimean family| ]] <!-- main article -->

[[Category:Temperament families]]

[[Category:Catalogs of rank-3 temperaments]]