Xenharmonic Wiki

Canou family: Difference between revisions

← Older edit
VisualWikitext
m Text replacement - "rank-3 temperament" to "rank-3 temperament"
Switch to Sintel's badness, WE & CWE tunings. - all 13-limit semicanou extensions (not very notable). - redundant category
 
Line 1: Line 1:
{{Technical data page}}
{{Technical data page}}
The '''canou family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[canousma]], 4802000/4782969 ({{monzo| 4 -14 3 4 }}), a 7-limit comma measuring about 6.9 [[cent]]s.
The '''canou family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[canousma]] ({{monzo|legend=1| 4 -14 3 4 }}, [[ratio]]: 4802000/4782969), a 7-limit comma measuring about 6.9 [[cent]]s.


== Canou ==
== Canou ==
{{Main| Canou temperament }}
{{Main| Canou }}


The canou temperament features a [[period]] of an [[octave]] and [[generator]]s of [[3/2]] and [[81/70]]. The ~81/70-generator is about 255 cents. Three make [[14/9]]; four make [[9/5]]. It therefore splits the large septimal diesis, [[49/48]], into three equal parts, making two distinct [[interseptimal interval]]s related to the 35th harmonic.  
The canou temperament features a [[period]] of an [[octave]] and [[generator]]s of [[3/2]] and [[81/70]]. The ~81/70 generator is about 255 cents wide, three of which make [[14/9]], and four make [[9/5]]. It therefore splits the large septimal diesis, [[49/48]], into three equal parts, guaranteeing the existence of two [[interseptimal interval]]s related to the 35th harmonic.  


A basic tuning option would be [[99edo]], although [[80edo]] is even simpler and distinctive. More intricate tunings are provided by [[311edo]] and [[410edo]], whereas the [[optimal patent val]] goes up to [[1131edo]], relating it to the [[amicable]] temperament.  
A basic tuning option would be [[99edo]], although [[80edo]] is even simpler and distinctive. More intricate tunings are provided by [[311edo]] and [[410edo]], whereas the [[optimal patent val]] goes up to [[1131edo]], associating it with the [[amicable]] temperament.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[4802000/4782969]]
[[Comma list]]: 4802000/4782969


{{Mapping|legend=1| 1 0 0 -1 | 0 1 2 2 | 0 0 -4 3 }}
{{Mapping|legend=1| 1 0 0 -1 | 0 1 2 2 | 0 0 -4 3 }}
: mapping generators: ~2, ~3, ~81/70
: mapping generators: ~2, ~3, ~81/70


Line 22: Line 21:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.3175, ~81/70 = 254.6220
* [[WE]]: ~2 = 1199.9597{{c}}, ~3/2 = 702.3492{{c}}, ~81/70 = 254.6168{{c}}
: [[error map]]: {{val| 0.0000 +0.3625 -0.1667 -0.3249 }}
: [[error map]]: {{val| -0.040 +0.354 -0.163 -0.317 }}
* [[CWE]]: ~2 = 1200.0000, ~3/2 = 702.3455, ~81/70 = 254.6237
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3455{{c}}, ~81/70 = 254.6237{{c}}
: error map: {{val| 0.0000 +0.3904 -0.1175 -0.2640 }}
: error map: {{val| 0.000 +0.390 -0.118 -0.264 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 35: Line 34:
{{Optimal ET sequence|legend=1| 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b }}
{{Optimal ET sequence|legend=1| 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b }}


[[Badness]] (Smith): 1.122 × 10<sup>-3</sup>
[[Badness]] (Sintel): 4.95


[[Complexity spectrum]]: 4/3, 9/7, 9/8, 7/6, 6/5, 10/9, 5/4, 8/7, 7/5
[[Complexity spectrum]]: 4/3, 9/7, 9/8, 7/6, 6/5, 10/9, 5/4, 8/7, 7/5


== Undecimal canou ==
== Undecimal canou ==
The fifth is in the range where a stack of four (i.e. a major third) can serve as ~[[19/15]] and a stack of five (i.e. a major seventh) can serve as ~[[19/10]], tempering out [[1216/1215]]. Moreover, the last generator of ~81/70 is sharpened to slightly overshoot [[22/19]], so it only makes sense to temper out their difference, [[1540/1539]]. The implied 11-limit comma is the [[symbiotic comma]], which suggests the [[wilschisma]] should also be tempered out in the 13-limit.  
The fifth is in the range where a stack of four (i.e. a major third) can serve as ~[[19/15]] and a stack of five (i.e. a major seventh) can serve as ~[[19/10]], tempering out [[1216/1215]]. Moreover, the last generator of ~81/70 is sharpened to slightly overshoot [[22/19]], so it only makes sense to temper out their difference, [[1540/1539]]. The implied 11-limit comma is the [[symbiotic comma]], which suggests the [[wilschisma]] should also be tempered out in the [[13-limit]].  


Since the syntonic comma has been split in two, it is natural to map [[19/17]] to the mean of [[9/8]] and [[10/9]], tempering out [[1445/1444]]. From a commatic point of view, notice the other 11-limit comma, [[42875/42768]], is {{nowrap| S34 × S35<sup>2</sup> }}, suggesting tempering out [[595/594]] (S34 × S35), [[1156/1155]] (S34), and [[1225/1224]] (S35), which coincides with above. Finally, we can map [[23/20]] to the fourth complement of 22/19 to make an equidistant sequence consisting of 7/6, 22/19, 23/20, and 8/7, tempering out [[760/759]]. 311edo remains an excellent tuning in all the limits.  
Since the syntonic comma has been split in two, it is natural to map [[19/17]] to the mean of [[9/8]] and [[10/9]], tempering out [[1445/1444]], while the other 11-limit comma, [[42875/42768]] (S34⋅S35<sup>2</sup>), suggests tempering out [[595/594]] (S34⋅S35), [[1156/1155]] (S34), and [[1225/1224]] (S35), which coincides with above. Finally, we can map [[23/20]] to the fourth complement of 22/19 to make an equidistant sequence consisting of 7/6, 22/19, 23/20, and 8/7, tempering out [[760/759]]. [[311edo]] remains an excellent tuning in all the limits.  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 51: Line 50:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.2115, ~81/70 = 254.6215
* [[WE]]: ~2 = 1200.0568{{c}}, ~3/2 = 702.2009{{c}}, ~81/70 = 254.6291{{c}}
: [[error map]]: {{val| 0.0000 +0.2565 -0.3768 -0.5383 +0.2980 }}
: [[error map]]: {{val| +0.057 +0.303 -0.314 -0.480 +0.201 }}
* [[CWE]]: ~2 = 1200.0000, ~3/2 = 702.1829, ~81/70 = 254.6186
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1829{{c}}, ~81/70 = 254.6186{{c}}
: error map: {{val| 0.0000 +0.2279 -0.4221 -0.6043 +0.1069 }}
: error map: {{val| 0.0000 +0.228 -0.422 -0.604 +0.107 }}


{{Optimal ET sequence|legend=1| 94, 99e, 118, 193, 212, 311, 740, 1051d }}
{{Optimal ET sequence|legend=1| 94, 99e, 118, 193, 212, 311, 740, 1051d }}


[[Badness]] (Smith): 2.04 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.45


[[Complexity spectrum]]: 4/3, 9/8, 9/7, 7/6, 5/4, 6/5, 10/9, 11/9, 8/7, 12/11, 11/10, 14/11, 11/8, 7/5
[[Complexity spectrum]]: 4/3, 9/8, 9/7, 7/6, 5/4, 6/5, 10/9, 11/9, 8/7, 12/11, 11/10, 14/11, 11/8, 7/5
Line 70: Line 69:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~3/2 = 702.2075, ~81/70 = 254.6183
* WE: ~2 = 1200.0501{{c}}, ~3/2 = 702.2100{{c}}, ~81/70 = 254.6345{{c}}
* CWE: ~2 = 1200.0000, ~3/2 = 702.1889, ~81/70 = 254.6222
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1889{{c}}, ~81/70 = 254.6222{{c}}


{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}


Badness (Smith): 2.56 × 10<sup>-3</sup>
Badness (Sintel): 2.39


=== 17-limit ===
=== 17-limit ===
Line 85: Line 84:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~3/2 = 702.2296, ~51/44 = 254.6012
* WE: ~2 = 1200.0630{{c}}, ~3/2 = 702.2317{{c}}, ~51/44 = 254.6224{{c}}
* CWE: ~2 = 1200.0000, ~3/2 = 702.2055, ~51/44 = 254.6066
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2055{{c}}, ~51/44 = 254.6066{{c}}


{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}


Badness (Smith): 1.49 × 10<sup>-3</sup>
Badness (Sintel): 1.41


=== 19-limit ===
=== 19-limit ===
Line 100: Line 99:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930
* WE: ~2 = 1200.0624{{c}}, ~3/2 = 702.2377{{c}}, ~22/19 = 254.6139{{c}}
* CWE: ~2 = 1200.0000, ~3/2 = 702.2117, ~22/19 = 254.5983
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2117{{c}}, ~22/19 = 254.5983{{c}}


{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}
{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}


Badness (Smith): 1.00 × 10<sup>-3</sup>
Badness (Sintel): 1.03


=== 23-limit ===
=== 23-limit ===
Line 115: Line 114:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~3/2 = 702.2361, ~22/19 = 254.6222
* WE: ~2 = 1200.0004{{c}}, ~3/2 = 702.2361{{c}}, ~22/19 = 254.6225{{c}}
* CWE: ~2 = 1200.0000, ~3/2 = 702.2359, ~22/19 = 254.6223
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2359{{c}}, ~22/19 = 254.6223{{c}}


{{Optimal ET sequence|legend=0| 94, 193f, 212gh, 217, 311 }}
{{Optimal ET sequence|legend=0| 94, 193f, 212gh, 217, 311 }}


Badness (Smith): 0.948 × 10<sup>-3</sup>
Badness (Sintel): 1.09


== Canta ==
== Canta ==
By adding [[896/891]], the pentacircle comma, [[33/32]] is equated with 28/27, so the scale is filled with this 33/32~28/27 mixture. This may be described as 75e & 80 & 99e, and 80edo makes the optimal. It has a natural extension to the 13-limit since 896/891 = (352/351)(364/363), named ''gentcanta'' in earlier materials.  
By adding [[896/891]], the pentacircle comma, [[33/32]] is equated with 28/27, so the scale is filled with this 33/32~28/27 mixture. This may be described as {{nowrap| 75e & 80 & 99e }}, and 80edo makes the optimal. It has a natural extension to the 13-limit since 896/891 = (352/351)⋅(364/363), named ''gentcanta'' in earlier materials.  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 132: Line 131:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.8093, ~64/55 = 254.3378
* [[WE]]: ~2 = 1199.0708{{c}}, ~3/2 = 703.1969{{c}}, ~64/55 = 254.4161{{c}}
: [[error map]]: {{val| 0.0000 +0.8543 +1.9537 -0.1940 +6.0769 }}
: [[error map]]: {{val| -0.929 +0.313 +0.557 -0.113 +1.820 }}
* [[CWE]]: ~2 = 1200.0000, ~3/2 = 703.5249, ~64/55 = 254.5492
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.5249{{c}}, ~64/55 = 254.5492{{c}}
: error map: {{val| 0.0000 +1.5699 +2.5393 +1.8714 +5.2799 }}
: error map: {{val| 0.000 +1.570 +2.539 +1.871 +5.280 }}


{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e, 457bcddeeee }}
{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e, 457bcddeeee }}


[[Badness]] (Smith): 4.52 × 10<sup>-3</sup>
[[Badness]] (Sintel): 5.43


=== 13-limit ===
=== 13-limit ===
Line 149: Line 148:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~3/2 = 703.6228, ~64/55 = 254.3447
* WE: ~2 = 1199.0093{{c}}, ~3/2 = 703.2884{{c}}, ~64/55 = 254.4219{{c}}
* CWE: ~2 = 1200.0000, ~3/2 = 703.8323, ~64/55 = 254.5887
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8323{{c}}, ~64/55 = 254.5887{{c}}


{{Optimal ET sequence|legend=0| 75e, 80, 99ef, 179ef }}
{{Optimal ET sequence|legend=0| 75e, 80, 99ef, 179ef, 462bccddeeeff }}


Badness (Smith): 4.78 × 10<sup>-3</sup>
Badness (Sintel): 4.47


== Semicanou ==
== Semicanou ==
Semicanou adds [[9801/9800]], the kalisma, to the comma list, and may be described as 80 & 94 & 118. It splits the octave into two equal parts, each representing 99/70~140/99. Note that {{nowrap| 99/70 {{=}} (81/70)(11/9) }}, this extension is more than natural.  
Semicanou adds 9801/9800, the [[kalisma]], to the comma list, and may be described as {{nowrap| 80 & 94 & 118 }}. It splits the octave into two equal parts, each representing 99/70~140/99. This takes advantage of the fact that {{nowrap| 99/70 {{=}} (81/70)⋅(11/9) }}.  


The other comma necessary to define it is 14641/14580, the [[semicanousma]], which is the difference between [[121/120]] and [[243/242]]. By flattening the 11th harmonic by about one cent, it identifies [[20/11]] by three [[11/9]]'s stacked, so an octave can be divided into 11/9, 11/9, 11/9, and 11/10.  
The other comma necessary to define it is 14641/14580, the [[semicanousma]], which is the difference between [[121/120]] and [[243/242]]. By flattening the 11th harmonic by about one cent, it identifies [[20/11]] by three [[11/9]]'s stacked, so an octave can be divided into 11/9, 11/9, 11/9, and 11/10.  
Line 166: Line 165:


{{Mapping|legend=1| 2 0 0 -2 1 | 0 1 2 2 2 | 0 0 -4 3 -1 }}
{{Mapping|legend=1| 2 0 0 -2 1 | 0 1 2 2 2 | 0 0 -4 3 -1 }}
: mapping generators: ~99/70, ~3, ~81/70
: mapping generators: ~99/70, ~3, ~81/70


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~99/70 = 600.0000, ~3/2 = 702.4262, ~81/70 = 254.6191
* [[WE]]: ~99/70 = 600.0142{{c}}, ~3/2 = 702.4017{{c}}, ~81/70 = 254.6228{{c}}
: [[error map]]: {{val| 0.0000 +0.4712 +0.0625 -0.1163 -1.0846 }}
: [[error map]]: {{val| +0.028 +0.475 +0.055 -0.126 -1.066 }}
* [[CWE]]: ~99/70 = 600.0000, ~3/2 = 702.4048, ~81/70 = 254.6179
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 702.4048{{c}}, ~81/70 = 254.6179{{c}}
: error map: {{val| 0.0000 +0.4498 +0.0245 -0.1627 -1.1262 }}
: error map: {{val| 0.0000 +0.450 +0.024 -0.163 -1.126 }}


{{Optimal ET sequence|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}
{{Optimal ET sequence|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}


[[Badness]] (Smith): 2.20 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.64
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 14641/14580
 
Mapping: {{mapping| 2 0 0 -2 1 -11 | 0 1 2 2 2 5 | 0 0 -4 3 -1 6 }}
 
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~3/2 = 702.4802, ~81/70 = 254.6526
* CWE: ~99/70 = 600.0000, ~3/2 = 702.4945, ~81/70 = 254.6511
 
{{Optimal ET sequence|legend=0| 80f, 94, 118f, 198, 410 }}
 
Badness (Smith): 2.97 × 10<sup>-3</sup>
 
<!-- debatable canonicity
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 715/714, 1089/1088, 1225/1224, 14641/14580
 
Mapping: {{mapping| 2 0 0 -2 1 -11 -10 | 0 1 2 2 2 5 6 | 0 0 -4 3 -1 6 -2 }}
 
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~3/2 = 702.4415, ~81/70 = 254.6663
 
{{Optimal ET sequence|legend=0| 94, 118f, 198g, 212g, 292, 410 }}
 
Badness (Smith): 2.42 × 10<sup>-3</sup>
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 715/714, 1089/1088, 1216/1215, 1225/1224, 1445/1444
 
Mapping: {{mapping| 2 0 0 -2 1 -11 -10 -12 | 0 1 2 2 2 5 6 7 | 0 0 -4 3 -1 6 -2 -4 }}
 
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~3/2 = 702.4030, ~81/70 = 254.6870
 
{{Optimal ET sequence|legend=0| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}
 
Badness (Smith): 2.18 × 10<sup>-3</sup>
-->
=== Semicanoumint ===
This extension was named ''semicanou'' in the earlier materials. It adds [[352/351]], the minthma, to the comma list, so that the flat ~11/9 simultaneously represents ~39/32.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 9801/9800, 14641/14580
 
Mapping: {{mapping| 2 0 0 -2 1 11 | 0 1 2 2 2 -1 | 0 0 -4 3 -1 -1 }}
 
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~3/2 = 702.5374, ~81/70 = 254.6819
* CTE: ~99/70 = 600.0000, ~3/2 = 702.7916, ~81/70 = 254.6704
 
{{Optimal ET sequence|legend=0| 80, 94, 118, 174d, 198, 490f }}
 
Badness (Smith): 2.70 × 10<sup>-3</sup>
 
=== Semicanouwolf ===
This extension was named ''gentsemicanou'' in the earlier materials. It adds [[351/350]], the ratwolfsma, as wells as [[364/363]], the gentle comma, to the comma list. Since 351/350 = (81/70)/(15/13), the 81/70-generator simultaneously represents 15/13, adding a lot of fun to the scale.
 
Not supported by many patent vals, 80edo easily makes the optimal. Yet 104edo in 104c val and 118edo in 118f val are worth mentioning, and the temperament may be described as 80 & 104c & 118f.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 364/363, 11011/10935
 
Mapping: {{mapping| 2 0 0 -2 1 0 | 0 1 2 2 2 3 | 0 0 -4 3 -1 -5 }}
 
Optimal tunings:
* CTE: ~55/39 = 600.0000, ~3/2 = 702.7417, ~15/13 = 254.3382
* CWE: ~55/39 = 600.0000, ~3/2 = 702.8092, ~15/13 = 254.3396
 
{{Optimal ET sequence|legend=0| 80, 104c, 118f, 198f, 420cff }}
 
Badness (Smith): 3.51 × 10<sup>-3</sup>


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Canou family| ]] <!-- main article -->
[[Category:Canou family| ]] <!-- main article -->
[[Category:Canou| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Canou_family"