Horwell temperaments: Difference between revisions

This is a collection of [[rank-2 temperament|rank-2]] '''horwell temperaments''', which temper out the [[horwell comma]] ({{monzo|legend=1| -16 1 5 1 }}, [[ratio]]: 65625/65536).

* ''[[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 }}. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator. It is a restriction of [[24576/24565 #2.3.5.7.17 subgroup (prime archagall)|prime archagall]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 65625/65536, 420175/419904

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

[[Optimal tuning]]s:  

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

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

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

: ''For the 5-limit version, see [[Father–3 equivalence continuum #Mutt (5-limit)]].''

 

{{Mapping|legend=1| 3 -2 6 20 | 0 7 1 -12 }}

[[Optimal tuning]]s:

* [[WE]]: ~63/50 = 400.0351{{c}}, ~5/4 = 385.9974{{c}} (~126/125 = 14.0377{{c}})

: [[error map]]: {{val| +0.105 -0.043 -0.105 -0.092 }}

* [[CWE]]: ~63/50 = 400.0000{{c}}, ~5/4 = 385.9638{{c}} (~126/125 = 14.0362{{c}})

[[Badness]] (Sintel): 0.719

Mapping: {{mapping| 3 -2 6 20 21 | 0 7 1 -12 -11 }}

Optimal tunings:

* WE: ~44/35 = 399.9783{{c}}, ~5/4 = 385.9993{{c}} (~126/125 = 13.9790{{c}})

* CWE: ~44/35 = 400.0000{{c}}, ~5/4 = 386.0208{{c}} (~126/125 = 13.9792{{c}})

{{Optimal ET sequence|legend=0| 84, 87, 171, 258 }}

Badness (Sintel): 1.93

Mapping: {{mapping| 3 -2 6 20 21 14 | 0 7 1 -12 -11 -3 }}

Optimal tunings:

* WE: ~44/35 = 399.9610{{c}}, ~5/4 = 385.9842{{c}} (~126/125 = 13.9768{{c}})

* CWE: ~44/35 = 400.0000{{c}}, ~5/4 = 386.0231{{c}} (~126/125 = 13.9769{{c}})

{{Optimal ET sequence|legend=0| 84, 87, 171, 258, 429ef }}

Badness (Sintel): 1.20

== Oquatonic ==

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

Oquatonic has a period of 1/28 octave and tempers out the horwell (65625/65536) and the [[dimcomp comma]] (390625/388962). In this temperament, the [[5/4]] major third is mapped to 9\28.

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 }}

[[Optimal tuning]]s:

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

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

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

{{Optimal ET sequence|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}

[[Badness]] (Sintel): 2.23

Comma list: 1375/1372, 6250/6237, 65625/65536

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

Optimal tunings:

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

* CWE: ~128/125 = 42.8571{{c}}, ~3/2 = 702.0174{{c}}

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

Badness (Sintel): 1.58

Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197

 

Optimal tunings:  

* 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 }}

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Emka]].''

[[File:Scale Tree Graph For Emkay.png|thumb|Scale tree graph for emkay.]]

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.

[[Comma list]]: 65625/65536, 244140625/243045684

{{Mapping|legend=1| 1 -13 -2 39 | 0 27 8 -67 }}

[[Optimal tuning]]s:

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

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

{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}

[[Badness]] (Sintel): 3.43

Comma list: 3025/3024, 4000/3993, 65625/65536

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

Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197

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

== Kastro ==

: ''For the 5-limit version, see [[Very high accuracy temperaments #Astro]].''

Kastro may be described as the {{nowrap| 109 & 118 }} temperament, named by [[Petr Pařízek]] in 2011 as a variation of ''astro''<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.  

[[Comma list]]: 65625/65536, 117649/116640

: mapping generators: ~2, ~6272/3375

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.1529{{c}}, ~6272/3375 = 1067.9515{{c}}

: [[error map]]: {{val| +0.153 +0.567 +0.256 -1.749 }}

* [[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

Comma list: 385/384, 3388/3375, 12005/11979

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

Optimal tunings:

* WE: ~2 = 1200.2427{{c}}, ~224/121 = 1068.0296{{c}}

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

{{Optimal ET sequence|legend=0| 109, 118, 345de, 463de, 581dde }}

Badness (Sintel): 1.74

Comma list: 169/168, 364/363, 385/384, 3388/3375

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

Optimal tunings:

* WE: ~2 = 1200.4303{{c}}, ~13/7 = 1068.2040{{c}}

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

{{Optimal ET sequence|legend=0| 109, 118f, 227f }}

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}}

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

[[Badness]] (Sintel): 6.82

Optimal tunings:

* WE: ~45/44 = 37.5025{{c}}, ~3/2 = 701.5912{{c}}

* 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:

* WE: ~45/44 = 37.5021{{c}}, ~3/2 = 701.5769{{c}}

* 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

== References ==