Xenharmonic Wiki

Kalismic temperaments: Difference between revisions

← Older edit
VisualWikitext
Update keys
m Update
 
(8 intermediate revisions by 4 users not shown)
Line 1: Line 1:
__FORCETOC__
__FORCETOC__
These are rank-3 temperaments tempering out [[9801/9800]]. Temperaments discussed elsewhere are:  
{{Technical data page}}
These are [[rank-3 temperament]]s tempering out [[9801/9800]]. Temperaments discussed elsewhere are:  
* ''[[Jubilee]]'' (+50/49 or 99/98) → [[Jubilismic family #Jubilee|Jubilismic family]]
* ''[[Jubilee]]'' (+50/49 or 99/98) → [[Jubilismic family #Jubilee|Jubilismic family]]
* ''[[Fantastic]]'' (+225/224) → [[Marvel family #Fantastic|Marvel family]]
* ''[[Fantastic]]'' (+225/224) → [[Marvel family #Fantastic|Marvel family]]
* ''[[Bisector]]'' (+121/120 or 245/243) → [[Sensamagic family #Bisector|Sensamagic family]]
* ''[[Bisector]]'' (+121/120 or 245/243) → [[Sensamagic family #Bisector|Sensamagic family]]
* ''[[Julius]]'' or ''[[varda]]'' (+176/175) → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank three family]]
* ''[[Varda]]'' (+176/175) → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]
* ''[[Hagrid]]'' (+243/242) → [[Cataharry family #Hagrid|Cataharry family]]
* ''[[Hagrid]]'' (+243/242) → [[Cataharry family #Hagrid|Cataharry family]]
* ''[[Uniwiz]]'' (+385/384) → [[Keenanismic temperaments #Uniwiz|Keenanismic temperaments]]
* ''[[Uniwiz]]'' (+385/384) → [[Keenanismic temperaments #Uniwiz|Keenanismic temperaments]]
Line 15: Line 16:
* ''[[Semicanou]]'' (+14641/14580) → [[Canou family #Semicanou|Canou family]]
* ''[[Semicanou]]'' (+14641/14580) → [[Canou family #Semicanou|Canou family]]


Considered below are odin, loki, van gogh, rishi, hnoss, and gersemi, but for the rank-4 temperament, see [[Rank-4 temperament #Kalismic (9801/9800)]].
Considered below are odin, lycoris, van gogh, sif, loki, pessoal, and linus. For the rank-4 temperament, see [[Rank-4 temperament #Kalismic (9801/9800)]].


== Odin ==
== Odin ==
Line 75: Line 76:
[[Badness]]: 0.297 × 10<sup>-3</sup>
[[Badness]]: 0.297 × 10<sup>-3</sup>


== Hnoss ==
== Sif ==
To the wizma {{monzo| -6 -8 2 5 }} = 420175/419904, the kalisma is a natural complement, as their product is the [[tinge]].
[[Subgroup]]: 2.3.5.7.11


[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 9801/9800, 2097152/2096325


[[Comma list]]: 9801/9800, 41503/41472
{{Mapping|legend=1| 2 0 1 7 11 | 0 1 0 1 -1 | 0 0 4 -5 -1 }}


{{Mapping|legend=1| 2 0 1 2 6 | 0 1 4 0 2 | 0 0 -5 2 -3 }}
: mapping generators: ~99/70, ~3


: mapping generators: ~99/70, ~3, ~144/77
[[Optimal tuning]] ([[CTE]]): ~99/70 = 1\2, ~3/2 = 701.9492, ~48/35 = 546.6041


{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 176, 198, 248, 270, 342, 612, 954, 1566, 4086dee, 5652cddeee }}
{{Optimal ET sequence|legend=1| 22, 46, 68, 114, 134, 156, 178, 202, 224, 270, 494, 742, 764, 966, 1236, 1506, 2742, 3236, 3506, 4742d, 10990ccdde, 15732ccdddee }}


[[Badness]]: 0.368 × 10<sup>-3</sup>
[[Badness]]: 0.423 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 2080/2079, 17303/17280
Comma list: 4096/4095, 6656/6655, 9801/9800
 
Mapping: {{mapping| 2 0 1 2 6 -3 | 0 1 4 0 2 1 | 0 0 -5 2 -3 4 }}
 
{{Optimal ET sequence|legend=1| 22f, 32cf, 54cff, 72, 166, 198, 270, 634, 904, 1174, 1880ef }}
 
Badness: 0.867 × 10<sup>-3</sup>
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 715/714, 1089/1088, 1225/1224, 2025/2023
 
Mapping: {{mapping| 2 0 1 2 6 -3 0 | 0 1 4 0 2 1 6 | 0 0 -5 2 -3 4 -6 }}
 
{{Optimal ET sequence|legend=1| 22f, 54cffgg, 72, 166g, 198g, 270, 364, 436, 634g, 706f }}
 
Badness: 0.862 × 10<sup>-3</sup>
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 715/714, 1225/1224, 1540/1539, 2080/2079, 4200/4199
 
Mapping: {{mapping| 2 0 1 2 6 -3 0 13 | 0 1 4 0 2 1 6 2 | 0 0 -5 2 -3 4 -6 -6 }}
 
{{Optimal ET sequence|legend=1| 72, 94, 166g, 198g, 270, 436, 634g, 706f }}
 
Badness: 0.901 × 10<sup>-3</sup>
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 715/714, 1225/1224, 1540/1539, 2080/2079, 2530/2527, 2737/2736
 
Mapping: {{mapping| 2 0 1 2 6 -3 0 13 19 | 0 1 4 0 2 1 6 2 -2 | 0 0 -5 2 -3 4 -6 -6 -2 }}
 
{{Optimal ET sequence|legend=1| 72, 94, 166g, 270, 342f, 436, 706fi }}
 
Badness: 1.14 × 10<sup>-3</sup>
 
=== Gersemi ===
The extension to 13-limit with [[4225/4224]] is weak but facilitates the use of 18/7 as the equave. [[Fokker block]]s of 128 notes are available for the latter, corresponding to 94edo. 18/7 is split into 4 parts that become ~19/15 in 19-limit. Also, (18/7)<sup>3</sup> ~ 17/1 via the [[5832/5831|chlorisma]]. However, the tones 9/8 and (19/15)/(9/8) = 152/135 have distinct mappings.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 4225/4224, 9801/9800, 41503/41472


Mapping: {{mapping| 2 0 1 2 6 9 | 0 1 9 -2 5 -6 | 0 0 -10 4 -6 7 }}
Mapping: {{mapping| 2 0 1 7 11 16 | 0 1 0 1 -1 -3 | 0 0 4 -5 -1 1 }}


: mapping generators: ~99/70, ~3, ~154/65
Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.9902, ~48/35 = 546.6064


{{Optimal ET sequence|legend=1| 44, 50, 94, 144, 176, 220, 270, 590, 684, 954 }}
{{Optimal ET sequence|legend=1| 22, 46, 156f, 178, 202f, 224, 270, 494, 764, 1012, 1236, 1506, 3236, 3506, 4742df, 6248cdf, 8248cdef, 12990ccddeff, 14496ccdddeeff }}


Badness: 1.06 × 10<sup>-3</sup>
Badness: 0.339 × 10<sup>-3</sup>
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 1089/1088, 1225/1224, 2025/2023, 4225/4224
 
Mapping: {{mapping| 2 0 1 2 6 9 0 | 0 1 9 -2 5 -6 12 | 0 0 -10 4 -6 7 -12 }}
 
{{Optimal ET sequence|legend=1| 44, 50, 94, 144g, 176g, 220g, 270, 364, 414, 634g, 684 }}
 
Badness: 1.46 × 10<sup>-3</sup>
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 1089/1088, 1225/1224, 1729/1728, 2926/2925, 3762/3757
 
Mapping: {{mapping| 2 0 1 2 6 9 0 1 | 0 1 9 -2 5 -6 12 11 | 0 0 -10 4 -6 7 -12 -11 }}
 
{{Optimal ET sequence|legend=1| 44, 50, 94, 144gh, 176g, 220g, 270, 414h, 590, 634g, 684h }}
 
Badness: 1.11 × 10<sup>-3</sup>
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 897/896, 1089/1088, 1225/1224, 1729/1728, 2737/2736, 2926/2925
 
Mapping: {{mapping| 2 0 1 2 6 9 0 1 7 | 0 1 9 -2 5 -6 12 11 3 | 0 0 -10 4 -6 7 -12 -11 -3 }}
 
{{Optimal ET sequence|legend=1| 44, 50, 94, 144gh, 176g, 220g, 226, 270, 320i, 364i, 414hi }}
 
Badness: 1.23 × 10<sup>-3</sup>


== Loki ==
== Loki ==
Line 225: Line 147:
Badness: 0.391 × 10<sup>-3</sup>
Badness: 0.391 × 10<sup>-3</sup>


== Rishi ==
== Linus ==
The 7-limit comma {{monzo| 65 -84 10 16 }} ~ 0.13¢ has the ratio of the exponents of 3 and 2 that is close to the one in 81/8. The square root of the latter is close to 35/11. This suggests tempering out (81/8)(35/11)<sup>-2</sup>, which is the kalisma.
{{Main| Linus }}
 
Linus [[tempering out|tempers out]] the [[linus comma]], {{monzo| 11 -10 -10 10 }} in the 7-limit and can be described as the 80 & 130 & 270 temperament. It tempers out 9801/9800 and 391314/390625 in the 11-limit; 1001/1000, 4225/4224, and 4459/4455 in the 13-limit. The 1/10-octave period interval represents 15/14, three of which represents 16/13, and five of which represents both 99/70 and 140/99.


Apart from 35/11, 35/33, and the equivalents of their squares, 81/8 and 9/8, another equave that comes to mind is 3/2, especially after tempering out the [[chalmersia]]. When 3/2 is chosen as the equave, Fokker blocks of 34 notes can be used that are close to [[34edf]] and 58edo.
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 578509309952/576650390625
{{Mapping|legend=1| 10 0 0 -11 | 0 1 0 1 | 0 0 1 1 }}
: mapping generators: ~15/14, ~3, ~5
[[Optimal tuning]] ([[CTE]]): ~15/14 = 1\10, ~3/2 = 702.0437, ~5/4 = 386.5040
{{Optimal ET sequence|legend=1| 50, 60, 80, 130, 270, 1270, 1540, 1810, 1940, 2080, 2210c, 2480c }}
[[Badness]]: 1.12 × 10<sup>-3</sup>
=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 9801/9800, 572145834917888/571919811374025
[[Comma list]]: 9801/9800, 391314/390625


{{Mapping|legend=1| 2 0 3 -10 -4 | 0 1 2 4 4 | 0 0 8 -5 3 }}
{{Mapping|legend=1| 10 0 0 -11 4 | 0 1 0 1 -1 | 0 0 1 1 2 }}


: mapping generators: ~99/70, ~3, ~17364375/14172488
[[Optimal tuning]] ([[CTE]]): ~15/14 = 1\10, ~3/2 = 702.0010, ~5/4 = 386.5927


{{Optimal ET sequence|legend=1| 24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 7414, 9874, 12828e }}
{{Optimal ET sequence|legend=1| 50, 60e, 80, 130, 190, 270, 670, 940, 1130, 1400, 1800c, 2070c, 2340c }}


[[Badness]]: 2.10 × 10<sup>-3</sup>
[[Badness]]: 1.12 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 9801/9800, 10648/10647, 371293/371250
Comma list: 1001/1000, 4225/4224, 4459/4455
 
Mapping: {{mapping| 10 0 0 -11 4 37 | 0 1 0 1 -1 0 | 0 0 1 1 2 0 }}


Mapping: {{mapping| 2 0 3 -10 -4 2 | 0 1 2 4 4 3 | 0 0 8 -5 3 7 }}
Optimal tuning (CTE): ~15/14 = 1\10, ~3/2 = 702.0010, ~5/4 = 386.5928


{{Optimal ET sequence|legend=1| 24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 5414, 6920, 7414, 9874, 12828e }}
Optimal ET sequence: {{Optimal ET sequence| 50, 60e, 80, 130, 190, 270, 590, 730, 860, 1130, 1590df, 1860def }}


[[Badness]]: 0.505 × 10<sup>-3</sup>
Badness: 0.771 × 10<sup>-3</sup>


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Kalismic temperaments| ]] <!-- main article -->
[[Category:Kalismic temperaments| ]] <!-- main article -->
[[Category:Kalismic| ]] <!-- key article -->
[[Category:Kalismic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Kalismic_temperaments"