Horwell temperaments: Difference between revisions

* ''[[Countercata]]'' (+5120/5103) → [[Kleismic family #Countercata|Kleismic family]]

* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]

* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila|Mabila family]]

* ''[[Narayana]]'' (+321489/320000) → [[Vishnu family #Narayana|Vishnu family]]

* ''[[Maquiloid]]'' (+686/675) → [[Maquila family #Maquiloid|Maquila family]]

 

== Fifthplus ==

 

Fifthplus tempers out the [[wizma]] in addition to the horwell comma, and may be described as the {{nowrap| 22 & 171 }} temperament. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator.

Fifthplus may be extended to the 2.3.5.7.17 subgroup, called prime archagall, derived from the fact that it is an extension of [[archagall]] to a prime subgroup. In either the 7-limit or the add-17 subgroup, [[171edo]] is exceptionally efficient and accurate, but in the latter case 171edo is the tuning where [[2401/2400]] (S49), [[2500/2499]] (S50), and [[1701/1700]] (S18/S20) all vanish, which is natural because this temperament tempers out [[1225/1224]] (S35, S49⋅S50) and [[5832/5831]] ((S18/S20)/S49) while not tempering out any of above individually.  

[[Subgroup]]: 2.3.5.7

 

{{Mapping|legend=1| 1 -12 10 -22 | 0 23 -13 42 }}

: mapping generators: ~2, ~5488/3645

* [[WE]]: ~2 = 1200.0934{{c}}, ~5488/3645 = 708.8291{{c}}

: [[error map]]: {{val| +0.093 -0.007 -0.158 -0.059 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5488/3645 = 708.7752{{c}}

: error map: {{val| 0.000 -0.126 -0.391 -0.268 }}

 

{{Optimal ET sequence|legend=1| 22, 105d, 127d, 149, 171, 1903c, 2074c, …, 3613ccd }}

 

=== Prime archagall ===

 

 

 

 

Badness (Sintel): 0.421

 

 

Mutt tempers out the [[landscape comma]] in addition to the horwell comma, and may be described as the {{nowrap| 84 & 87 }} temperament.  

== Oquatonic ==

: ''For the 5-limit version, see [[28th-octave temperaments #Oquatonic (5-limit)]].''

 

The name ''oquatonic'' was given by [[Petr Pařízek]] in 2011 as an abbreviation of the Italian [[wiktionary: ottantaquatro|''ottantaquatro'' ("eighty-four")]]<ref name="petr's long post"/>.  

[[Comma list]]: 65625/65536, 390625/388962

{{Mapping|legend=1| 28 0 65 123 | 0 1 0 -1 }}

: mapping generators: ~128/125, ~3

* [[WE]]: ~128/125 = 42.8570{{c}}, ~3/2 = 702.1112{{c}}

: [[error map]]: {{val| -0.004 +0.152 -0.609 +0.477 }}

* [[CWE]]: ~128/125 = 42.8571{{c}}, ~3/2 = 702.1132{{c}}

: error map: {{val| 0.000 +0.158 -0.599 +0.489 }}

 

 

 

=== 11-limit ===

 

 

Mapping: {{mapping| 28 0 65 123 230 | 0 1 0 -1 -3 }}

 

* WE: ~128/125 = 42.8577{{c}}, ~3/2 = 702.0275{{c}}

 

 

 

=== 13-limit ===

 

 

Mapping: {{mapping| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}

 

* WE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0289{{c}}

* CWE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0288{{c}}

{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}

Badness (Sintel): 0.908

Emkay may be described as the {{nowrap| 87 & 224 }} temperament. It tempers out the same 5-limit comma as the [[emka]] (37 & 50), but with the horwell comma (65625/65536) rather than the hemimean comma (3136/3125) tempered out.

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0279{{c}}, ~4536/3125 = 648.2405{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~4536/3125 = 648.2254{{c}}

: error map: {{val| 0.000 +0.133 -0.510 +0.069 }}

Mapping: {{mapping| 1 -13 -2 39 4 | 0 27 8 -67 -1 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 648.2254{{c}}

{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535 }}

Badness (Sintel): 1.18

Mapping: {{mapping| 1 -13 -2 39 4 1 | 0 27 8 -67 -1 5 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 648.2251{{c}}

{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535 }}

Badness (Sintel): 0.738

{{Mapping|legend=1| 1 -26 13 -23 | 0 31 -12 29 }}

[[Optimal tuning]]s:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~6272/3375 = 1067.8174{{c}}

: error map: {{val| 0.000 +0.384 -0.122 -2.122 }}

{{Optimal ET sequence|legend=1| 109, 118, 345d, 463d, 581dd }}

[[Badness]] (Sintel): 4.64

Mapping: {{mapping| 1 -26 13 -23 -9 | 0 31 -12 29 14 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~224/121 = 1067.8166{{c}}

Badness (Sintel): 1.74

Mapping: {{mapping| 1 -26 13 -23 -9 -23 | 0 31 -12 29 14 30 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1067.8267{{c}}

Badness (Sintel): 1.93

Bezique splits the octave into 32 equal parts and reaches 3/2, 8/5 and 11/8 in just one generator with the 64-tone mos. A notable edo tuning overshadowed by [[224edo]] is [[320edo]]. Bezique was named by [[Eliora]] in 2023 for the fact that the card game of bezique is played with two packs of 32 cards.  

[[Optimal tuning]]s:

* [[WE]]: ~100352/98415 = 37.5038{{c}}, ~3/2 = 701.6058{{c}}

: error map: {{val| 0.000 -0.401 -0.368 -0.108 }}

{{Optimal ET sequence|legend=1| 96d, 224, 544, 768, 1312, 2080bc }}

[[Badness]] (Sintel): 6.82

Optimal tunings:

* CWE: ~45/44 = 37.5000{{c}}, ~3/2 = 701.5566{{c}}

{{Optimal ET sequence|legend=0| 96d, 224, 544, 768 }}

Badness (Sintel): 2.25

Optimal tunings:

* CWE: ~45/44 = 37.5000{{c}}, ~3/2 = 701.5490{{c}}

{{Optimal ET sequence|legend=0| 96d, 224, 544, 768, 1312 }}

Badness (Sintel): 1.23