Breed family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,046 edits
m Restore the category of the corresponding temp
 
(35 intermediate revisions by 7 users not shown)
Line 1: Line 1:
{{Technical data page}}
{{Technical data page}}
The '''breed family''' of [[temperament]]s are [[Rank-3 temperament|rank-3]] [[microtemperament]]s which [[Tempering out|temper out]] [[2401/2400]]. While it is so accurate it hardly matters what is used to temper it, or whether it is tempered at all, the [[optimal patent val]] 2749et will certainly do the trick.  
The '''breed family''' of [[rank-3 temperament]]s are [[microtemperament]]s which [[tempering out|temper out]] [[2401/2400]].  


== Breed ==
== Breed ==
Breed has the same lattice structure as the [[2.5.7 subgroup]] of JI. In terms of extension from the 5-limit, it splits 25/6 into four equal parts representing 10/7. While it is so accurate it hardly matters what is used to temper it, or whether it is tempered at all, the [[optimal patent val]] [[2749edo|2749et]] will certainly do the trick.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 8: Line 10:


{{Mapping|legend=1| 1 1 1 2 | 0 2 1 1 | 0 0 2 1 }}
{{Mapping|legend=1| 1 1 1 2 | 0 2 1 1 | 0 0 2 1 }}
 
: Mapping generators: ~2, ~49/40, ~10/7
: mapping generators: ~2, ~49/40, ~10/7


[[Mapping to lattice]]: [{{val| 0 2 -1 0 }}, {{val| 0 0 -2 -1 }}]
[[Mapping to lattice]]: [{{val| 0 2 -1 0 }}, {{val| 0 0 -2 -1 }}]
Line 16: Line 17:
: 49/40 length = 0.7858, 8/7 length = 1.1241
: 49/40 length = 0.7858, 8/7 length = 1.1241
: Angle (49/40, 8/7) = 107.367°
: Angle (49/40, 8/7) = 107.367°
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0206{{c}}, ~49/40 = 350.9724{{c}}, ~10/7 = 617.6826{{c}}
: [[error map]]: {{val| +0.021 +0.010 +0.044 -0.130 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9743{{c}}, ~10/7 = 617.6807{{c}}
: error map: {{val| 0.000 -0.006 +0.022 -0.171 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.5
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.5
 
[[Optimal tuning]] (POTE): ~2 = 1\1, ~49/40 = 350.9664, ~10/7 = 617.6720


{{Optimal ET sequence|legend=1| 27, 31, 41, 58, 68, 72, 99, 171, 441, 612, 1966, 2308, 2578, 2749, 3361d }}
{{Optimal ET sequence|legend=1| 27, 31, 41, 58, 68, 72, 99, 171, 441, 612, 1966, 2308, 2578, 2749, 3361d }}


[[Badness]]: 0.0153 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.0674


[[Projection pair]]: 3 = ~2401/800 to 2.5.7
[[Projection pair]]: <code>3 2401/800</code> to 2.5.7


Scales: [[breed11]]
Scales: [[breed11]]
Line 36: Line 41:
{{Main| Jove }}
{{Main| Jove }}


Jove (formerly known as wonder) tempers out 243/242 and 441/440. Jove converts breed into an 11-limit temperament via 441/440, which equates 49/40 with 11/9, and 243/242, which tells us 11/9 can serve as a neutral third. While jove is no longer a super-accurate microtemperament like breed, it has the advantage of adjusting its tuning to deal with the 11-limit. 72, 130, 171 and 202 are good edos for jove.
Jove (formerly known as wonder) tempers out [[243/242]], [[441/440]], and [[540/539]]. Jove converts breed into an 11-limit temperament via 441/440, which equates [[49/40]] with [[11/9]], and 243/242, which tells us 11/9 can serve as a neutral third (as in exactly half of [[~]][[3/2]]). While jove is no longer a super-accurate microtemperament like breed, it has the advantage of adjusting its tuning to deal with the 11-limit. [[72edo|72]], [[130edo|130]], [[171edo|171]] and [[202edo|202]] are good edo tunings for jove.


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 43: Line 48:


{{Mapping|legend=1| 1 1 1 2 2 | 0 2 1 1 5 | 0 0 2 1 0 }}
{{Mapping|legend=1| 1 1 1 2 2 | 0 2 1 1 5 | 0 0 2 1 0 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0985{{c}}, ~11/9 = 350.5322{{c}}, ~10/7 = 617.8782{{c}}
: [[error map]]: {{val| +0.098 -0.792 +0.073 -0.218 +1.540 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.5374{{c}}, ~10/7 = 617.8712{{c}}
: error map: {{val| 0.000 -0.880 -0.034 -0.417 +1.369 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[11-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11
* [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11
 
[[Optimal tuning]] (POTE): ~2 = 1\1, ~11/9 = 350.5034, ~10/7 = 617.8275


{{Optimal ET sequence|legend=1| 27e, 31, 41, 58, 72, 130, 202 }}
{{Optimal ET sequence|legend=1| 27e, 31, 41, 58, 72, 130, 202 }}


[[Badness]]: 0.241 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.290


Projection pairs: ~3 = ~242/81, ~5 = ~2200/441, ~7 = ~440/63, ~11 = ~644204/59049 to 2.7/5.11/9
[[Projection pair]]s: <code>3 242/81, 5 2200/441, 7 440/63, 11 644204/59049</code> to 2.7/5.11/9


=== Jovial ===
=== Jovial ===
Line 60: Line 69:
Comma list: 243/242, 364/363, 441/440
Comma list: 243/242, 364/363, 441/440


Mapping: {{mapping| 1 1 1 2 2 1 | 0 2 1 1 5 11 | 0 0 2 1 0 -1 }}
{{Mapping|legend=0| 1 1 1 2 2 1 | 0 2 1 1 5 11 | 0 0 2 1 0 -1 }}
 
Optimal tunings:
* WE: ~2 = 1199.9775{{c}}, ~11/9 = 350.7113{{c}}, ~10/7 = 617.8170{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.7130{{c}}, ~10/7 = 617.8178{{c}}


Minimax tuning:
Minimax tuning:
Line 66: Line 79:
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/5.15/13
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/5.15/13


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.7179, ~10/7 = 617.8286
{{Optimal ET sequence|legend=0| 27eff, 31f, 41, 58, 72, 130, 243, 301e, 373e }}


Optimal ET sequence: {{Optimal ET sequence| 27eff, 31f, 41, 58, 72, 130, 243, 301e, 373e }}
Badness (Sintel): 0.583
 
Badness: 0.624 × 10<sup>-3</sup>


==== 17-limit ====
==== 17-limit ====
Line 77: Line 88:
Comma list: 243/242, 364/363, 441/440, 595/594
Comma list: 243/242, 364/363, 441/440, 595/594


Mapping: {{mapping| 1 1 1 2 2 1 3 | 0 2 1 1 5 11 9 | 0 0 2 1 0 -1 -3 }}
{{Mapping|legend=0| 1 1 1 2 2 1 3 | 0 2 1 1 5 11 9 | 0 0 2 1 0 -1 -3 }}
 
Optimal tunings:
* WE: ~2 = 1200.0876{{c}}, ~11/9 = 350.7203{{c}}, ~10/7 = 617.5766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.7138{{c}}, ~10/7 = 617.5610{{c}}


Minimax tuning:
Minimax tuning:
* 17-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.17/9
* 17-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.17/9


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.6947, ~10/7 = 617.5315
{{Optimal ET sequence|legend=0| 27effg, 31fg, 41, 58, 72, 130, 171, 243 }}
 
Optimal ET sequence: {{Optimal ET sequence| 27effg, 31fg, 41, 58, 72, 130, 171, 243 }}


Badness: 0.741 × 10<sup>-3</sup>
Badness (Sintel): 0.704


==== Heartlandia ====
==== Heartlandia ====
Line 93: Line 106:
Comma list: 243/242, 364/363, 441/440, 1452/1445
Comma list: 243/242, 364/363, 441/440, 1452/1445


Mapping: {{mapping| 1 1 1 2 2 1 3 | 0 4 0 1 10 23 12 | 0 0 2 1 0 -1 -1 }}
{{Mapping|legend=0| 1 1 1 2 2 1 3 | 0 4 0 1 10 23 12 | 0 0 2 1 0 -1 -1 }}
: Mapping generators: ~2, ~119/108, ~27/17


: mapping generators: ~2, ~119/108, ~27/17
Optimal tunings:
* WE: ~2 = 1199.5594{{c}}, ~119/108 = 175.3533{{c}}, ~27/17 = 793.6847{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~119/108 = 175.3707{{c}}, ~27/17 = 793.7559{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~119/108 = 175.4177, ~27/17 = 793.9762
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41, 75ce, 89, 116cefg, 130g }}


Optimal ET sequence: {{Optimal ET sequence| 14cf, 27effg, 41, 89, 130g }}
Badness (Sintel): 2.84


===== 19-limit =====
===== 19-limit =====
Line 106: Line 122:
Comma list: 171/170, 243/242, 324/323, 364/363, 441/440
Comma list: 171/170, 243/242, 324/323, 364/363, 441/440


Mapping: {{mapping| 1 1 1 2 2 1 3 3 | 0 4 0 1 10 23 12 4 | 0 0 2 1 0 -1 -1 1 }}
{{Mapping|legend=0| 1 1 1 2 2 1 3 3 | 0 4 0 1 10 23 12 4 | 0 0 2 1 0 -1 -1 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.3862, ~19/12 = 793.9558
Optimal tunings:
* WE: ~2 = 1199.7333{{c}}, ~21/19 = 175.3472{{c}}, ~19/12 = 793.7794{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3584{{c}}, ~19/12 = 793.8194{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14cf, 27effg, 41, 89, 130g }}
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41, 75ce, 89, 116cefg, 130g }}
 
Badness (Sintel): 2.29
 
=== Jovis ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 351/350, 441/440
 
{{Mapping|legend=0| 1 1 1 2 2 2 | 0 2 1 1 5 -3 | 0 0 2 1 0 5 }}
 
Optimal tunings:
* WE: ~2 = 1200.1435{{c}}, ~11/9 = 350.4354{{c}}, ~10/7 = 618.1775{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4402{{c}}, ~10/7 = 618.1759{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 31, 45ef, 58, 72, 103, 130, 233, 363 }}
 
Badness (Sintel): 0.507


=== Jofur ===
=== Jofur ===
Line 117: Line 152:
Comma list: 144/143, 196/195, 243/242
Comma list: 144/143, 196/195, 243/242


Mapping: {{mapping| 1 1 1 2 2 4 | 0 2 1 1 5 -1 | 0 0 2 1 0 0 }}
{{Mapping|legend=0| 1 1 1 2 2 4 | 0 2 1 1 5 -1 | 0 0 2 1 0 0 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.4477, ~10/7 = 618.8891
Optimal tunings:
* WE: ~2 = 1198.9534{{c}}, ~11/9 = 351.1412{{c}}, ~10/7 = 618.3494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.2536{{c}}, ~10/7 = 618.5932{{c}}


Optimal ET sequence: {{Optimal ET sequence| 27e, 31, 41, 58, 99ef, 157eff }}
{{Optimal ET sequence|legend=0| 27e, 31, 41, 58, 99ef, 157eff }}


Badness: 0.749 × 10<sup>-3</sup>
Badness (Sintel): 0.701


=== Jovis ===
=== Semijove ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 243/242, 351/350, 441/440
Comma list: 243/242, 441/440, 676/675


Mapping: {{mapping| 1 1 1 2 2 2 | 0 2 1 1 5 -3 | 0 0 2 1 0 5 }}
{{Mapping|legend=0| 2 0 1 3 -1 -1 | 0 2 1 1 5 4 | 0 0 2 1 0 2 }}
: mapping generators: ~55/39, ~26/15, ~10/7


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.3935, ~10/7 = 618.1036
Optimal tunings:
* WE: ~55/39 = 599.9886{{c}}, ~26/15 = 950.6645{{c}}, ~10/7 = 618.1458{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~26/15 = 950.6755{{c}}, ~10/7 = 618.1489{{c}}


Optimal ET sequence: {{Optimal ET sequence| 27e, 31, 45ef, 58, 72, 103, 130, 233, 363 }}
{{Optimal ET sequence|legend=0| 48c, 58, 72, 130 }}


Badness: 0.542 × 10<sup>-3</sup>
Badness (Sintel): 0.948


== Agni ==
== Agni ==
Agni tempers out [[385/384]] and [[1375/1372]], so that a stack of three 10/7 generators is identified with [[16/11]] plus an octave. [[140edo]], [[212edo]] and [[284edo]] show how it can be tuned distinctly from [[#Jove|jove]].
This temperament suggests at least two extensions to the 13-limit: pavaka (31 & 68 & 72), named after an alias of Agni, tempers out [[625/624]]; and svaha (41 & 68 & 72), named after Agni's wife, tempers out [[325/324]]. The edos above can be used to tune both, which reduce them to [[quadritikleismic]] without much additional damage, but note pavaka is a detemperament of [[benediction]] whereas svaha is a detemperament of [[manna]].
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 150: Line 194:
: 49/40 length = 0.756, 10/7 length = 0.819
: 49/40 length = 0.756, 10/7 length = 0.819
: Angle (49/40, 10/7) = 106.460 degrees
: Angle (49/40, 10/7) = 106.460 degrees
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4168{{c}}, ~49/40 = 350.8363{{c}}, ~10/7 = 617.2187{{c}}
: [[error map]]: {{val| +0.417 +0.134 -0.623 +0.063 -0.890 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.8369{{c}}, ~10/7 = 616.9907{{c}}
: error map: {{val| 0.000 -0.281 -1.495 -0.998 -2.290 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 23/10 3/10 2/5 0 -2/5 }}, {{monzo| 12/5 2/5 1/5 0 -1/5 }}, {{monzo| 23/10 3/10 -3/5 0 3/5 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 23/10 3/10 2/5 0 -2/5 }}, {{monzo| 12/5 2/5 1/5 0 -1/5 }}, {{monzo| 23/10 3/10 -3/5 0 3/5 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/5
 
{{Optimal ET sequence|legend=1| 31, 41, 68, 72, 140, 171e, 212, 284, 496ce, 527cee, 739cdeee, 811ccdeee, 1023ccdeee }}
 
[[Badness]] (Sintel): 0.593
 
=== Pavaka ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 385/384, 625/624, 1375/1372
 
{{Mapping|legend=0| 1 1 1 2 5 -1 | 0 2 1 1 0 2 | 0 0 2 1 -3 8 }}


[[Optimal tuning]] (POTE): ~2 = 1\1, ~49/40 = 350.7145, ~10/7 = 617.0044
Optimal tunings:
* WE: ~2 = 1200.4413{{c}}, ~49/40 = 350.8092{{c}}, ~10/7 = 617.3718{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 350.8046{{c}}, ~10/7 = 617.1572{{c}}


{{Optimal ET sequence|legend=1| 27, 31, 41, 68, 72, 140, 171e, 212, 284, 496ce, 527cee, 739cdeee, 811ccdeee, 1023ccdeee }}
{{Optimal ET sequence|legend=0| 31, 68, 72, 103, 140, 212, 243e, 315ef, 455eef, 770cdeeeff }}


[[Badness]]: 0.494 × 10<sup>-3</sup>
Badness (Sintel): 0.863


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


Comma list: 385/384, 625/624, 1375/1372
Comma list: 325/324, 385/384, 1375/1372


Mapping: {{mapping| 1 1 1 2 5 -1 | 0 2 1 1 0 2 | 0 0 2 1 -3 8 }}
{{Mapping|legend=0| 1 1 1 2 5 4 | 0 2 1 1 0 6 | 0 0 2 1 -3 -4 }}


Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 350.6803, ~10/7 = 617.1448
Optimal tunings:
* WE: ~2 = 1200.3582{{c}}, ~49/40 = 351.0784{{c}}, ~10/7 = 617.0054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0491{{c}}, ~10/7 = 616.8317{{c}}


Optimal ET sequence: {{Optimal ET sequence| 31, 68, 72, 103, 140, 212, 243e, 315ef, 455eef, 770cdeeeff }}
{{Optimal ET sequence|legend=0| 41, 68, 72, 140, 212, 393ce, 465cef, 677cdeeff }}


Badness: 0.923 × 10<sup>-3</sup>
Badness (Sintel): 0.997


== Zisa ==
== Zisa ==
Line 181: Line 246:


{{Mapping|legend=1| 1 1 1 2 -3 | 0 2 1 1 8 | 0 0 2 1 8 }}
{{Mapping|legend=1| 1 1 1 2 -3 | 0 2 1 1 8 | 0 0 2 1 8 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9178{{c}}, ~49/40 = 350.0519{{c}}, ~10/7 = 617.8150{{c}}
: [[error map]]: {{val| -0.082 +0.067 +0.286 -0.123 -0.135 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.0615{{c}}, ~10/7 = 617.8571{{c}}
: error map: {{val| 0.000 +0.168 +0.462 +0.093 +0.032 }}


{{Optimal ET sequence|legend=1| 31, 68e, 99e, 130, 239, 270, 670, 940, 1210, 2150c }}
{{Optimal ET sequence|legend=1| 31, 68e, 99e, 130, 239, 270, 670, 940, 1210, 2150c }}


[[Badness]]: 0.640 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.769


=== 13-limit ===
=== 13-limit ===
Line 191: Line 262:
Comma list: 1001/1000, 1716/1715, 4096/4095
Comma list: 1001/1000, 1716/1715, 4096/4095


Mapping: {{mapping| 1 1 1 2 -3 7 | 0 2 1 1 8 -6 | 0 0 2 1 8 -3 }}
{{Mapping|legend=0| 1 1 1 2 -3 7 | 0 2 1 1 8 -6 | 0 0 2 1 8 -3 }}


{{Optimal ET sequence|legend=1| 31, 78f, 99e, 109, 130, 239, 270, 571, 701, 841, 971, 1241 }}
Optimal tunings:
* WE: ~2 = 1199.9787{{c}}, ~49/40 = 350.9976{{c}}, ~10/7 = 617.8701{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0039{{c}}, ~10/7 = 617.8795{{c}}


Badness: 0.830 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 31, 78f, 99e, 109, 130, 239, 270, 571, 701, 841, 971, 1241 }}
 
Badness (Sintel): 0.776


== Lif ==
== Lif ==
Lif tempers out the [[olympia]] and the [[lifthrasirsma]], so that the [[rastma]] is split in halves with each part standing in not only for [[441/440]][[~]][[540/539]], but the [[32805/32768|schisma]]. Lif readily extends to the [[13-limit]] and the no-17 [[19-limit]], where the [[513/512|undevicesimal schisma]] is also equated with the schisma.
This temperament was named by [[Flora Canou]] in 2023 along with the [[weak restriction]], [[lifthrasir]], and the corresponding comma, lifthrasirsma, after the human survivors of Ragnarök.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 204: Line 283:
{{Mapping|legend=1| 1 1 1 2 8 | 0 2 1 1 -12 | 0 0 2 1 -2 }}
{{Mapping|legend=1| 1 1 1 2 8 | 0 2 1 1 -12 | 0 0 2 1 -2 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~49/40 = 351.0959, ~10/7 = 617.6652
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9386{{c}}, ~49/40 = 351.0586{{c}}, ~10/7 = 617.7065{{c}}
: [[error map]]: {{val| -0.061 +0.101 +0.096 -0.184 +0.075 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.0806{{c}}, ~10/7 = 617.7228{{c}}
: error map: {{val| 0.000 +0.206 +0.213 -0.022 +0.269 }}


{{Optimal ET sequence|legend=1| 41, 89, 130, 229, 270, 581, 670, 711, 981, 1251, 2232e }}
{{Optimal ET sequence|legend=1| 41, 89, 130, 229, 270, 581, 670, 711, 981, 1251, 2232e }}


[[Badness]]: 0.793 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.953


=== 13-limit ===
=== 13-limit ===
Line 215: Line 298:
Comma list: 2080/2079, 2401/2400, 4096/4095
Comma list: 2080/2079, 2401/2400, 4096/4095


Mapping: {{mapping| 1 1 1 2 8 7 | 0 2 1 1 -12 -6 | 0 0 2 1 -2 -3 }}
{{Mapping|legend=0| 1 1 1 2 8 7 | 0 2 1 1 -12 -6 | 0 0 2 1 -2 -3 }}


Optimal tuning (CTE): ~2 = 1\1, ~49/40 = 351.0960, ~10/7 = 617.6533
Optimal tunings:
* WE: ~2 = 1199.9597{{c}}, ~49/40 = 351.0718{{c}}, ~10/7 = 617.6543{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0856{{c}}, ~10/7 = 617.6715{{c}}


{{Optimal ET sequence|legend=1| 41, 89, 99, 130, 270, 581, 711, 981, 1292, 1562 }}
{{Optimal ET sequence|legend=0| 41, 89, 99, 130, 270, 581, 711, 981, 1292, 1562 }}


Badness: 0.579 × 10<sup>-3</sup>
Badness (Sintel): 0.542


=== 2.3.5.7.11.13.19 subgroup ===
=== 2.3.5.7.11.13.19 subgroup ===
Line 228: Line 313:
Comma list: 1216/1215, 1729/1728, 2080/2079, 2401/2400
Comma list: 1216/1215, 1729/1728, 2080/2079, 2401/2400


Sval mapping: {{mapping| 1 1 1 2 8 7 0 | 0 2 1 1 -12 -6 11 | 0 0 2 1 -2 -3 2 }}
{{Mapping|legend=2| 1 1 1 2 8 7 0 | 0 2 1 1 -12 -6 11 | 0 0 2 1 -2 -3 2 }}
 
Optimal tunings:
* WE: ~2 = 1199.9752{{c}}, ~49/40 = 351.0908{{c}}, ~10/7 = 617.6473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0984{{c}}, ~10/7 = 617.6588{{c}}
 
{{Optimal ET sequence|legend=0| 41, 89, 130, 229, 270, 581, 851, 1562, 1832, 2413 }}
 
Badness (Sintel): 0.488
 
== Freyr ==
Freyr tempers out the [[symbiotic comma]], so that the [[rastma]] is split in halves with each part standing in not only for [[441/440]][[~]][[540/539]], but the [[5120/5103|aberschisma]]. Freyr easily extends to the [[13-limit]], equating the [[352/351|major minthma]] with the aberschisma, and the no-17 [[19-limit]], equating the [[513/512|undevicesimal schisma]] with it too.
 
It can be notated with [[neutral chain-of-fifths notation]] with two additional sets of accidentals, one for the generic comma step, and the other for the generic aberschisma step.
 
This temperament was named by [[Flora Canou]] in 2026, under the impression that it functions as a counterpart of [[#Freya|freya]].
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 2401/2400, 19712/19683
 
{{Mapping|legend=1| 1 1 1 2 -1 | 0 2 1 1 17 | 0 0 2 1 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9474{{c}}, ~49/40 = 351.1064{{c}}, ~10/7 = 617.6483{{c}}
: [[error map]]: {{val| -0.053 +0.205 +0.037 -0.176 -0.104 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.1156{{c}}, ~10/7 = 617.6499{{c}}
: error map: {{val| 0.000 +0.276 +0.102 -0.060 -0.003 }}
 
{{Optimal ET sequence|legend=1| 41, 58, 99e, 171e, 212, 270, 581, 851, 1121, 1972, 2881bc }}
 
[[Badness]] (Sintel): 1.23
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 2401/2400, 10648/10647
 
{{Mapping|legend=0| 1 1 1 2 -1 -2 | 0 2 1 1 17 23 | 0 0 2 1 -1 -2 }}


Optimal tuning (CTE): ~2 = 1\1, ~49/40 = 351.1007, ~10/7 = 617.6501
Optimal tunings:
* WE: ~2 = 1199.9432{{c}}, ~49/40 = 351.1128{{c}}, ~10/7 = 617.6419{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1232{{c}}, ~10/7 = 617.6431{{c}}


{{Optimal ET sequence|legend=1| 41, 89, 130, 229, 270, 581, 851, 1562, 1832, 2413 }}
{{Optimal ET sequence|legend=0| 41, 58, 99ef, 154cef, 171ef, 212, 270, 581, 851, 1490, 1760, 2071be, 2341b }}


Badness: 0.499 × 10<sup>-3</sup>
Badness (Sintel): 0.760
 
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1216/1215, 1540/1539, 2080/2079, 2401/2400
 
{{Mapping|legend=0| 1 1 1 2 -1 -2 0 | 0 2 1 1 17 23 11 | 0 0 2 1 -1 -2 2 }}
 
Optimal tunings:
* WE: ~2 = 1199.9435{{c}}, ~49/40 = 351.1127{{c}}, ~10/7 = 617.6404{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1236{{c}}, ~10/7 = 617.6459{{c}}
 
{{Optimal ET sequence|legend=0| 41, 58h, 99ef, 171ef, 212h, 270, 581, 851, 1490h, 1760h, 2071beh, 2341bh }}
 
Badness (Sintel): 0.523


== Baldur ==
== Baldur ==
Line 242: Line 382:


{{Mapping|legend=1| 2 0 1 3 7 | 0 2 1 1 -2 | 0 0 2 1 3 }}
{{Mapping|legend=1| 2 0 1 3 7 | 0 2 1 1 -2 | 0 0 2 1 3 }}
: Mapping generators: ~99/70, ~343/198, ~10/7


: mapping generators: ~99/70, ~343/198, ~10/7
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 600.0149{{c}}, ~343/198 = 950.9717{{c}}, ~10/7 = 617.7007{{c}}
: [[error map]]: {{val| +0.030 -0.012 +0.074 -0.109 -0.055 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~343/198 = 950.9565{{c}}, ~10/7 = 617.7017{{c}}
: error map: {{val| 0.000 -0.042 +0.046 -0.168 -0.126 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 3/4 0 1/2 1/2 -1/2 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 23/16 0 5/8 1/8 -1/8 }}, {{monzo| 23/16 0 5/8 -7/8 7/8 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 3/4 0 1/2 1/2 -1/2 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 23/16 0 5/8 1/8 -1/8 }}, {{monzo| 23/16 0 5/8 -7/8 7/8 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7


{{Optimal ET sequence|legend=1| 58, 72, 130, 198, 212, 270, 342, 612, 954, 1084, 1354, 1696, 4004de, 5700de }}
{{Optimal ET sequence|legend=1| 58, 72, 130, 198, 212, 270, 342, 612, 954, 1084, 1354, 1696, 4004de, 5700de }}


[[Badness]]: 0.166 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.200


[[Projection pair]]s: 2 9801/4900 3 117649/39204 7 9801/1400 11 913517247483640899/83082326424002500 to 5.7/2.99/4
[[Projection pair]]s: <code>2 9801/4900, 3 117649/39204, 7 9801/1400, 11 913517247483640899/83082326424002500</code> to 5.7/2.99/4


=== Greenland ===
=== Greenland ===
Line 261: Line 406:
Comma list: 676/675, 1001/1000, 1716/1715
Comma list: 676/675, 1001/1000, 1716/1715


Mapping: {{mapping| 2 0 1 3 7 -1 | 0 2 1 1 -2 4 | 0 0 2 1 3 2 }}
{{Mapping|legend=0| 2 0 1 3 7 -1 | 0 2 1 1 -2 4 | 0 0 2 1 3 2 }}


{{Optimal ET sequence|legend=1| 58, 72, 130, 198, 270, 940, 1210f }}
Optimal tunings:
* WE: ~99/70 = 599.9868{{c}}, ~26/15 = 951.0538{{c}}, ~10/7 = 617.8062{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~26/15 = 951.0689{{c}}, ~10/7 = 617.8067{{c}}


Badness: 0.433 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 58, 72, 130, 198, 270, 940, 1210f }}


Complexity spectrum: 15/13, 7/5, 8/7, 7/6, 4/3, 15/14, 5/4, 18/13, 13/12, 14/13, 13/10, 6/5, 16/15, 11/10, 9/7, 9/8, 16/13, 10/9, 14/11, 11/8, 15/11, 12/11, 13/11, 11/9
Badness (Sintel): 0.415


Projection pairs: 2 19600/9801 3 676/225 5 10400/2079 7 20384000/2910897 11 19208000000000000/1750211597736459 13 5026736/385875 to 10/7.200/99.26/15
Projection pairs: <code>2 19600/9801, 3 676/225, 5 10400/2079, 7 20384000/2910897, 11 19208000000000000/1750211597736459, 13 5026736/385875</code> to 10/7.200/99.26/15


==== Midnatssol ====
==== Midnatssol ====
Line 276: Line 423:
Comma list: 289/288, 442/441, 561/560, 676/675
Comma list: 289/288, 442/441, 561/560, 676/675


Mapping: {{mapping| 2 0 1 3 7 -1 5 | 0 2 1 1 -2 4 2 | 0 0 2 1 3 2 0 }}
{{Mapping|legend=0| 2 0 1 3 7 -1 5 | 0 2 1 1 -2 4 2 | 0 0 2 1 3 2 0 }}


===== Kompasretning =====
Optimal tunings:
Subgroup: 2.3.5.7.11.13.17.19
* WE: ~17/12 = 600.1118{{c}}, ~26/15 = 951.1907{{c}}, ~10/7 = 617.6100{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~26/15 = 951.0626{{c}}, ~10/7 = 617.5842{{c}}


Comma list: 289/288, 442/441, 561/560, 676/675, 1088/1083
{{Optimal ET sequence|legend=0| 58, 72, 130, 140, 198, 212g, 270g, 342fg, 482fgg }}


Mapping: {{mapping| 4 0 0 5 11 -4 10 17 | 0 2 1 1 -2 4 2 0 | 0 0 2 1 3 2 0 0 }}
Badness (Sintel): 0.720


== Freya ==
== Freya ==
Freya tempers out [[3025/3024]] and [[41503/41472]], so it splits breed's [[28/27]] into two [[55/54]]~[[56/55]]'s and [[8/7]] into two [[77/72]]'s. It also takes the [[225/224]][[~]][[1029/1024]] [[kleisma #As an interval region|kleisma]] to represent [[243/242]], and splits it in half to get [[385/384]], [[441/440]], and [[540/539]]. Since these are the commas tempered out by [[miracle]], the simple relations between them make freya an elegant miracle detemperament.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 2401/2400, 3025/3024
[[Comma list]]: 2401/2400, 3025/3024


{{Mapping|legend=1| 1 1 3 3 2 | 0 2 3 2 1 | 0 0 -4 -2 3 }}
{{Mapping|legend=1| 1 1 -1 1 5 | 0 2 3 2 1 | 0 0 4 2 -3 }}
: Mapping generators: ~2, ~49/40, ~84/55


: mapping generators: ~2, ~49/40, ~55/42
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0491{{c}}, ~49/40 = 350.9820{{c}}, ~84/55 = 733.3514{{c}}
: [[error map]]: {{val| +0.030 -0.012 +0.074 -0.109 -0.055 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9890{{c}}, ~84/55 = 733.3160{{c}}
: error map: {{val| 0.000 -0.042 +0.046 -0.168 -0.126 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[11-odd-limit]] eigenmonzos (unchanged-intervals): 2, 14/11, 4/3
* [[11-odd-limit]] [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3.11/7


{{Optimal ET sequence|legend=1| 31, 41, 72, 167, 188, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee }}
{{Optimal ET sequence|legend=1| 31, 41, 72, 167, 188, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee }}


[[Badness]]: 0.170 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.204


[[Projection pair]]s: ~3 = ~2401/800, ~5 = ~22880495169/4575312500, ~7 = ~1058841/151250, ~11 = ~33275/3024 to 2.49/5.77/3
[[Projection pair]]s: <code>3 2401/800, 5 22880495169/4575312500, 7 1058841/151250, 11 33275/3024</code> to 2.49/5.77/3


=== 13-limit ===
=== 13-limit ===
Tridecimal freya maps [[352/351]] to the half-kleisma. It has [[miraculous]] as a sub-temperament.
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 2401/2400, 3025/3024, 4096/4095
Comma list: 2401/2400, 3025/3024, 4096/4095


Mapping: {{mapping| 1 1 3 3 2 4 | 0 2 3 2 1 -9 | 0 0 -4 -2 3 6 }}
{{Mapping|legend=0| 1 1 -1 1 5 10 | 0 2 3 2 1 -9 | 0 0 4 2 -3 -6 }}


{{Optimal ET sequence|legend=1| 31, 41, 72f, 198f, 229, 239, 270, 571, 581, 851, 882, 1152, 1463, 1733, 2615 }}
Optimal tunings:
* WE: ~2 = 1199.9870{{c}}, ~49/40 = 351.0502{{c}}, ~84/55 = 733.2969{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0534{{c}}, ~84/55 = 733.3055{{c}}


Badness: 0.855 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 31, 41, 72f, 198f, 229, 239, 270, 571, 581, 851, 882, 1152, 1463, 1733, 2615 }}


Projection pairs: ~3 = ~2401/800, ~5 = ~22880495169/4575312500, ~7 = ~1058841/151250, ~11 = ~33275/3024, ~13 = ~1814078464000000000000000/139662717676432916098329 to 2.49/5.77/3
Badness (Sintel): 0.800
 
Projection pairs: <code>3 2401/800, 5 22880495169/4575312500, 7 1058841/151250, 11 33275/3024, 13 1814078464000000000000000/139662717676432916098329</code> to 2.49/5.77/3


=== Eir ===
=== Eir ===
[[User:VIxen|VIxen]] named this extension after a healer goddess or valkyrie from the Norse mythology, as it is an extension of freya with the ibnsinma that evokes associations with medicine.  
Eir maps [[325/324]], [[364/363]], and [[729/728]] to the half-kleisma and has [[manna]] as a sub-temperament. {{U|VIxen}} named this extension after a healer goddess or valkyrie from the Norse mythology, as it extends freya via the [[ibnsinma]], which evokes associations with medicine.  


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 323: Line 484:
Comma list: 2080/2079, 2401/2400, 3025/3024
Comma list: 2080/2079, 2401/2400, 3025/3024


Mapping: {{mapping| 1 1 3 3 2 0 | 0 2 3 2 1 6 | 0 0 -4 -2 3 5 }}
{{Mapping|legend=0| 1 1 -1 1 5 5 | 0 2 3 2 1 6 | 0 0 4 2 -3 -5 }}
 
{{Optimal ET sequence|legend=1| 13cdf, 31f, 41, 72, 157, 185cf, 198, 270, 581, 851, 1504, 1774f, 2085, 2355f }}


Badness: 0.581 × 10<sup>-3</sup>
Optimal tunings:
* WE: ~2 = 1199.9994{{c}}, ~49/40 = 351.0880{{c}}, ~84/55 = 733.2478{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0880{{c}}, ~84/55 = 733.2482{{c}}


==== 17-limit ====
{{Optimal ET sequence|legend=0| 41, 72, 157, 185cf, 198, 270, 581, 851, 1504, 1774f, 2085, 2355f }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 1225/1224, 2058/2057, 2080/2079, 2401/2400
Badness (Sintel): 0.543


Mapping: {{mapping| 1 1 3 3 2 0 7 | 0 2 3 2 1 6 6 | 0 0 -4 -2 3 5 -12 }}
==== 2.3.5.7.11.13.19 subgroup ====
 
Subgroup: 2.3.5.7.11.13.19
{{Optimal ET sequence|legend=1| 41g, 72, 198g, 239f, 270, 311, 509, 581, 1234, 1815 }}
 
Badness: 0.700 × 10<sup>-3</sup>
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 1225/1224, 1540/1539, 2080/2079, 3136/3135, 4200/4199
Comma list: 1540/1539, 2080/2079, 2401/2400, 3025/3024


Mapping: {{mapping| 1 1 3 3 2 0 7 6 | 0 2 3 2 1 6 6 -2 | 0 0 -4 -2 3 5 -12 -3 }}
{{Mapping|legend=2| 1 1 -1 1 5 5 3 | 0 2 3 2 1 6 -2 | 0 0 4 2 -3 -5 3 }}


Mapping generators: ~2, ~49/40, ~55/42
Optimal tunings:
* WE: ~2 = 1199.9951{{c}}, ~49/40 = 351.0906{{c}}, ~84/55 = 733.2475{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0913{{c}}, ~84/55 = 733.2473{{c}}


{{Optimal ET sequence|legend=1| 41g, 72, 198g, 239f, 270, 311, 581, 1234, 1815 }}
{{Optimal ET sequence|legend=0| 41, 72, 113, 157, 198, 270, 581, 851, 2085, 2355f }}


Badness: 0.692 × 10<sup>-3</sup>
Badness (Sintel): 0.500


=== Heimlaug ===
=== Heimlaug ===
VIxen named this extension after a völva (seeress) from the Gull-Þóris saga of Icelanders. It is an extension of freya with the [[1001/1000|fairytale comma]] and the [[936/935|ainisma]], both adding to the mystical theme. The one of prophecy is bolstered by that this extension has [[benediction]] as a subtemperament.  
Heimlaug maps [[351/350]] and [[625/624]] to the half-kleisma and has [[benediction]] as a subtemperament. VIxen named this extension after a völva (seeress) from the Gull-Þóris saga of Icelanders, as it extends freya via the [[1001/1000|fairytale comma]], adding to the mystical theme.  


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 360: Line 516:
Comma list: 1001/1000, 1716/1715, 3025/3024
Comma list: 1001/1000, 1716/1715, 3025/3024


Mapping: {{mapping| 1 1 3 3 2 7 | 0 2 3 2 1 6 | 0 0 -4 -2 3 -13 }}
{{Mapping|legend=0| 1 1 -1 1 5 -6 | 0 2 3 2 1 6 | 0 0 4 2 -3 13 }}
 
{{Optimal ET sequence|legend=1| 8bcef, 15bbccdeeff, 23bcf, 31, 72, 103, 167, 198, 270, 571, 643, 913f }}
 
Badness: 0.601 × 10<sup>-3</sup>


==== 17-limit ====
Optimal tunings:
Equave 10/7 and 16-note scales with that period are of interest.
* WE: ~2 = 1200.0601{{c}}, ~49/40 = 350.9562{{c}}, ~84/55 = 733.4471{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 350.9646{{c}}, ~84/55 = 733.4049{{c}}


Subgroup: 2.3.5.7.11.13.17
{{Optimal ET sequence|legend=0| 31, 72, 103, 167, 198, 270, 571, 643, 913f }}


Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 1716/1715
Badness (Sintel): 0.562
 
Mapping: {{mapping| 1 1 3 3 2 7 7 | 0 2 3 2 1 6 6 | 0 0 -4 -2 3 -13 -12 }}
 
: mapping generators: ~2, ~49/40, ~17/13
 
{{Optimal ET sequence|legend=1| 8bcefg, 15bbccdeeffggg, 23bcfg, 31, 64be, 72, 103, 167, 198g, 239, 270 }}
 
Badness: 0.829 × 10<sup>-3</sup>


== Vili ==
== Vili ==
Line 386: Line 531:
[[Comma list]]: 2401/2400, 391314/390625
[[Comma list]]: 2401/2400, 391314/390625


{{Mapping|legend=1| 1 1 5 4 10 | 0 2 3 2 6 | 0 0 -6 -3 -14 }}
{{Mapping|legend=1| 1 1 -1 1 -4 | 0 2 3 2 6 | 0 0 6 3 14 }}
: Mapping generators: ~2, ~49/40, ~175/132
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9872{{c}}, ~49/40 = 350.9740{{c}}, ~175/132 = 488.9418{{c}}
: [[error map]]: {{val| -0.013 -0.020 +0.272 -0.065 -0.238 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9728{{c}}, ~175/132 = 488.9475{{c}}
: error map: {{val| 0.000 -0.009 +0.290 -0.038 -0.216 }}


{{Optimal ET sequence|legend=1| 27e, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 670, 913, 1043, 1313, 1583 }}
{{Optimal ET sequence|legend=1| 27e, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 670, 913, 1043, 1313, 1583 }}


[[Badness]]: 1.26 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.51


=== 13-limit ===
=== 13-limit ===
Line 397: Line 549:
Comma list: 1001/1000, 1716/1715, 4225/4224
Comma list: 1001/1000, 1716/1715, 4225/4224


Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 3 2 6 1 | 0 0 -6 -3 -14 -1 }}
{{Mapping|legend=0| 1 1 -1 1 -4 3 | 0 2 3 2 6 1 | 0 0 6 3 14 1 }}


{{Optimal ET sequence|legend=1| 27e, 37, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 913f }}
Optimal tunings:
* WE: ~2 = 1200.0597{{c}}, ~49/40 = 350.9419{{c}}, ~65/49 = 488.9741{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 350.9441{{c}}, ~65/49 = 488.9475{{c}}


Badness: 0.738 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 27e, 37, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 913f }}
 
Badness (Sintel): 0.690


== Frigg ==
== Frigg ==
Line 408: Line 564:
[[Comma list]]: 2401/2400, 644204/643125
[[Comma list]]: 2401/2400, 644204/643125


{{Mapping|legend=1| 1 1 3 3 4 | 0 2 3 2 4 | 0 0 -10 -5 -11 }}
{{Mapping|legend=1| 1 1 -7 -2 -7 | 0 2 3 2 4 | 0 0 10 5 11 }}
: Mapping generators: ~2, ~49/40, ~88/49
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0183{{c}}, ~49/40 = 350.9854{{c}}, ~88/49 = 1013.3709{{c}}
: [[error map]]: {{val| +0.018 +0.034 +0.223 -0.037 -0.425 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9871{{c}}, ~88/49 = 1013.3556{{c}}
: error map: {{val| 0.000 +0.019 +0.203 -0.074 -0.458 }}


{{Optimal ET sequence|legend=1| 45e, 58, 103, 161, 212, 270, 643, 913, 1183e }}
{{Optimal ET sequence|legend=1| 45e, 58, 103, 161, 212, 270, 643, 913, 1183e }}


[[Badness]]: 1.79 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.15


=== 13-limit ===
=== 13-limit ===
Line 419: Line 582:
Comma list: 1001/1000, 1716/1715, 10648/10647
Comma list: 1001/1000, 1716/1715, 10648/10647


Mapping: {{mapping| 1 1 3 3 4 5 | 0 2 3 2 4 3 | 0 0 -10 -5 -11 -14 }}
{{Mapping|legend=0| 1 1 -7 -2 -7 -9 | 0 2 3 2 4 3 | 0 0 10 5 11 14 }}


{{Optimal ET sequence|legend=1| 45ef, 58, 103, 161, 212, 270, 643, 913f, 1614ef }} <nowiki>*</nowiki>
Optimal tunings:
* WE: ~2 = 1200.0387{{c}}, ~49/40 = 350.9502{{c}}, ~70/39 = 1013.4065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 350.9526{{c}}, ~70/39 = 1013.3747{{c}}


<nowiki>*</nowiki> [[optimal patent val]]: [[1241edo|1241]]
{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161, 212, 270, 643, 913f, 1614ef }} *


Badness: 0.934 × 10<sup>-3</sup>
<nowiki/>* optimal patent val: [[1241edo|1241]]
 
Badness (Sintel): 0.873


== Ennealimmic ==
== Ennealimmic ==
{{Distinguish|Ennealimnic}}
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 433: Line 602:


{{Mapping|legend=1| 9 1 1 12 0 | 0 2 3 2 0 | 0 0 0 0 1 }}
{{Mapping|legend=1| 9 1 1 12 0 | 0 2 3 2 0 | 0 0 0 0 1 }}
: Mapping generators: ~27/25, ~5/3, ~11


: mapping generators: ~27/25, ~5/3, ~11
[[Optimal tuning]]s:
* [[WE]]: ~27/25 = 133.3357{{c}}, ~5/3 = 884.3288{{c}}, ~11/8 = 551.2529{{c}}
: [[error map]]: {{val| +0.021 +0.038 +0.009 -0.139 -0.000 }}
* [[CWE]]: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3215{{c}}, ~11/8 = 551.2775{{c}}
: error map: {{val| 0.000 +0.021 -0.016 -0.183 -0.040 }}


{{Optimal ET sequence|legend=1| 27, 45, 72, 171, 198, 270, 342, 612, 954, 1323, 1395, 1665, 2007, 2277, 2619, 4284d, 6561dd }}
{{Optimal ET sequence|legend=1| 72, 171(e), 198, 270, 342, 612, 954, 1323, 1395, 1665, 2007, 2277, 2619, 4284d, 6561dd, 6903dd }}


[[Badness]]: 0.275 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.330


=== 13-limit ===
=== 13-limit ===
Line 445: Line 619:
Comma list: 2080/2079, 2401/2400, 4375/4374
Comma list: 2080/2079, 2401/2400, 4375/4374


Mapping: {{mapping| 9 1 1 12 0 -31 | 0 2 3 2 0 5 | 0 0 0 0 1 1 }}
{{Mapping|legend=0| 9 1 1 12 0 -31 | 0 2 3 2 0 5 | 0 0 0 0 1 1 }}


{{Optimal ET sequence|legend=1| 27, 45f, 54cff, 72, 171, 198, 270, 639, 711, 981, 1692e }}
Optimal tunings:
* WE: ~27/25 = 133.3276{{c}}, ~5/3 = 884.3660{{c}}, ~11/8 = 551.7221{{c}}
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3905{{c}}, ~11/8 = 551.7085{{c}}


Badness: 0.755 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 72, 171(ef), 198, 270, 639, 711, 981, 1692e }}
 
Badness (Sintel): 0.706


[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Breed family| ]] <!-- main article -->
[[Category:Breed family| ]] <!-- main article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Temperament families]]
[[Category:Catalogs of rank-3 temperaments]]

Latest revision as of 12:01, 19 October 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The breed family of rank-3 temperaments are microtemperaments which temper out 2401/2400.

Breed

Breed has the same lattice structure as the 2.5.7 subgroup of JI. In terms of extension from the 5-limit, it splits 25/6 into four equal parts representing 10/7. While it is so accurate it hardly matters what is used to temper it, or whether it is tempered at all, the optimal patent val 2749et will certainly do the trick.

Subgroup: 2.3.5.7

Comma list: 2401/2400

Mapping: [⟨1 1 1 2], ⟨0 2 1 1], ⟨0 0 2 1]]

Mapping generators: ~2, ~49/40, ~10/7

Mapping to lattice: [⟨0 2 -1 0], ⟨0 0 -2 -1]]

Lattice basis:

49/40 length = 0.7858, 8/7 length = 1.1241
Angle (49/40, 8/7) = 107.367°

Optimal tunings:

  • WE: ~2 = 1200.0206 ¢, ~49/40 = 350.9724 ¢, ~10/7 = 617.6826 ¢
error map: ⟨+0.021 +0.010 +0.044 -0.130]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9743 ¢, ~10/7 = 617.6807 ¢
error map: ⟨0.000 -0.006 +0.022 -0.171]

Minimax tuning:

Optimal ET sequence: 27, 31, 41, 58, 68, 72, 99, 171, 441, 612, 1966, 2308, 2578, 2749, 3361d

Badness (Sintel): 0.0674

Projection pair: 3 2401/800 to 2.5.7

Scales: breed11

Music

Jove

Main article: Jove

Jove (formerly known as wonder) tempers out 243/242, 441/440, and 540/539. Jove converts breed into an 11-limit temperament via 441/440, which equates 49/40 with 11/9, and 243/242, which tells us 11/9 can serve as a neutral third (as in exactly half of ~3/2). While jove is no longer a super-accurate microtemperament like breed, it has the advantage of adjusting its tuning to deal with the 11-limit. 72, 130, 171 and 202 are good edo tunings for jove.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440

Mapping: [⟨1 1 1 2 2], ⟨0 2 1 1 5], ⟨0 0 2 1 0]]

Optimal tunings:

  • WE: ~2 = 1200.0985 ¢, ~11/9 = 350.5322 ¢, ~10/7 = 617.8782 ¢
error map: ⟨+0.098 -0.792 +0.073 -0.218 +1.540]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.5374 ¢, ~10/7 = 617.8712 ¢
error map: ⟨0.000 -0.880 -0.034 -0.417 +1.369]

Minimax tuning:

Optimal ET sequence: 27e, 31, 41, 58, 72, 130, 202

Badness (Sintel): 0.290

Projection pairs: 3 242/81, 5 2200/441, 7 440/63, 11 644204/59049 to 2.7/5.11/9

Jovial

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440

Mapping: [⟨1 1 1 2 2 1], ⟨0 2 1 1 5 11], ⟨0 0 2 1 0 -1]]

Optimal tunings:

  • WE: ~2 = 1199.9775 ¢, ~11/9 = 350.7113 ¢, ~10/7 = 617.8170 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.7130 ¢, ~10/7 = 617.8178 ¢

Minimax tuning:

  • 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9
  • 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/5.15/13

Optimal ET sequence: 27eff, 31f, 41, 58, 72, 130, 243, 301e, 373e

Badness (Sintel): 0.583

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 441/440, 595/594

Mapping: [⟨1 1 1 2 2 1 3], ⟨0 2 1 1 5 11 9], ⟨0 0 2 1 0 -1 -3]]

Optimal tunings:

  • WE: ~2 = 1200.0876 ¢, ~11/9 = 350.7203 ¢, ~10/7 = 617.5766 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.7138 ¢, ~10/7 = 617.5610 ¢

Minimax tuning:

  • 17-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.17/9

Optimal ET sequence: 27effg, 31fg, 41, 58, 72, 130, 171, 243

Badness (Sintel): 0.704

Heartlandia

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 441/440, 1452/1445

Mapping: [⟨1 1 1 2 2 1 3], ⟨0 4 0 1 10 23 12], ⟨0 0 2 1 0 -1 -1]]

Mapping generators: ~2, ~119/108, ~27/17

Optimal tunings:

  • WE: ~2 = 1199.5594 ¢, ~119/108 = 175.3533 ¢, ~27/17 = 793.6847 ¢
  • CWE: ~2 = 1200.0000 ¢, ~119/108 = 175.3707 ¢, ~27/17 = 793.7559 ¢

Optimal ET sequence: 14cf, 27effg, 41, 75ce, 89, 116cefg, 130g

Badness (Sintel): 2.84

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 171/170, 243/242, 324/323, 364/363, 441/440

Mapping: [⟨1 1 1 2 2 1 3 3], ⟨0 4 0 1 10 23 12 4], ⟨0 0 2 1 0 -1 -1 1]]

Optimal tunings:

  • WE: ~2 = 1199.7333 ¢, ~21/19 = 175.3472 ¢, ~19/12 = 793.7794 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.3584 ¢, ~19/12 = 793.8194 ¢

Optimal ET sequence: 14cf, 27effg, 41, 75ce, 89, 116cefg, 130g

Badness (Sintel): 2.29

Jovis

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 441/440

Mapping: [⟨1 1 1 2 2 2], ⟨0 2 1 1 5 -3], ⟨0 0 2 1 0 5]]

Optimal tunings:

  • WE: ~2 = 1200.1435 ¢, ~11/9 = 350.4354 ¢, ~10/7 = 618.1775 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.4402 ¢, ~10/7 = 618.1759 ¢

Optimal ET sequence: 27e, 31, 45ef, 58, 72, 103, 130, 233, 363

Badness (Sintel): 0.507

Jofur

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 243/242

Mapping: [⟨1 1 1 2 2 4], ⟨0 2 1 1 5 -1], ⟨0 0 2 1 0 0]]

Optimal tunings:

  • WE: ~2 = 1198.9534 ¢, ~11/9 = 351.1412 ¢, ~10/7 = 618.3494 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.2536 ¢, ~10/7 = 618.5932 ¢

Optimal ET sequence: 27e, 31, 41, 58, 99ef, 157eff

Badness (Sintel): 0.701

Semijove

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 441/440, 676/675

Mapping: [⟨2 0 1 3 -1 -1], ⟨0 2 1 1 5 4], ⟨0 0 2 1 0 2]]

mapping generators: ~55/39, ~26/15, ~10/7

Optimal tunings:

  • WE: ~55/39 = 599.9886 ¢, ~26/15 = 950.6645 ¢, ~10/7 = 618.1458 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~26/15 = 950.6755 ¢, ~10/7 = 618.1489 ¢

Optimal ET sequence: 48c, 58, 72, 130

Badness (Sintel): 0.948

Agni

Agni tempers out 385/384 and 1375/1372, so that a stack of three 10/7 generators is identified with 16/11 plus an octave. 140edo, 212edo and 284edo show how it can be tuned distinctly from jove.

This temperament suggests at least two extensions to the 13-limit: pavaka (31 & 68 & 72), named after an alias of Agni, tempers out 625/624; and svaha (41 & 68 & 72), named after Agni's wife, tempers out 325/324. The edos above can be used to tune both, which reduce them to quadritikleismic without much additional damage, but note pavaka is a detemperament of benediction whereas svaha is a detemperament of manna.

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372

Mapping: [⟨1 1 1 2 5], ⟨0 2 1 1 0], ⟨0 0 2 1 -3]]

Mapping to lattice: [⟨0 2 1 1 0], ⟨0 0 2 1 -3]]

Lattice basis:

49/40 length = 0.756, 10/7 length = 0.819
Angle (49/40, 10/7) = 106.460 degrees

Optimal tunings:

  • WE: ~2 = 1200.4168 ¢, ~49/40 = 350.8363 ¢, ~10/7 = 617.2187 ¢
error map: ⟨+0.417 +0.134 -0.623 +0.063 -0.890]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.8369 ¢, ~10/7 = 616.9907 ¢
error map: ⟨0.000 -0.281 -1.495 -0.998 -2.290]

Minimax tuning:

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [23/10 3/10 2/5 0 -2/5⟩, [12/5 2/5 1/5 0 -1/5⟩, [23/10 3/10 -3/5 0 3/5⟩]
unchanged-interval (eigenmonzo) basis: 2.3.11/5

Optimal ET sequence: 31, 41, 68, 72, 140, 171e, 212, 284, 496ce, 527cee, 739cdeee, 811ccdeee, 1023ccdeee

Badness (Sintel): 0.593

Pavaka

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624, 1375/1372

Mapping: [⟨1 1 1 2 5 -1], ⟨0 2 1 1 0 2], ⟨0 0 2 1 -3 8]]

Optimal tunings:

  • WE: ~2 = 1200.4413 ¢, ~49/40 = 350.8092 ¢, ~10/7 = 617.3718 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.8046 ¢, ~10/7 = 617.1572 ¢

Optimal ET sequence: 31, 68, 72, 103, 140, 212, 243e, 315ef, 455eef, 770cdeeeff

Badness (Sintel): 0.863

Svaha

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1375/1372

Mapping: [⟨1 1 1 2 5 4], ⟨0 2 1 1 0 6], ⟨0 0 2 1 -3 -4]]

Optimal tunings:

  • WE: ~2 = 1200.3582 ¢, ~49/40 = 351.0784 ¢, ~10/7 = 617.0054 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0491 ¢, ~10/7 = 616.8317 ¢

Optimal ET sequence: 41, 68, 72, 140, 212, 393ce, 465cef, 677cdeeff

Badness (Sintel): 0.997

Zisa

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 5632/5625

Mapping: [⟨1 1 1 2 -3], ⟨0 2 1 1 8], ⟨0 0 2 1 8]]

Optimal tunings:

  • WE: ~2 = 1199.9178 ¢, ~49/40 = 350.0519 ¢, ~10/7 = 617.8150 ¢
error map: ⟨-0.082 +0.067 +0.286 -0.123 -0.135]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.0615 ¢, ~10/7 = 617.8571 ¢
error map: ⟨0.000 +0.168 +0.462 +0.093 +0.032]

Optimal ET sequence: 31, 68e, 99e, 130, 239, 270, 670, 940, 1210, 2150c

Badness (Sintel): 0.769

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 4096/4095

Mapping: [⟨1 1 1 2 -3 7], ⟨0 2 1 1 8 -6], ⟨0 0 2 1 8 -3]]

Optimal tunings:

  • WE: ~2 = 1199.9787 ¢, ~49/40 = 350.9976 ¢, ~10/7 = 617.8701 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0039 ¢, ~10/7 = 617.8795 ¢

Optimal ET sequence: 31, 78f, 99e, 109, 130, 239, 270, 571, 701, 841, 971, 1241

Badness (Sintel): 0.776

Lif

Lif tempers out the olympia and the lifthrasirsma, so that the rastma is split in halves with each part standing in not only for 441/440~540/539, but the schisma. Lif readily extends to the 13-limit and the no-17 19-limit, where the undevicesimal schisma is also equated with the schisma.

This temperament was named by Flora Canou in 2023 along with the weak restriction, lifthrasir, and the corresponding comma, lifthrasirsma, after the human survivors of Ragnarök.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 131072/130977

Mapping: [⟨1 1 1 2 8], ⟨0 2 1 1 -12], ⟨0 0 2 1 -2]]

Optimal tunings:

  • WE: ~2 = 1199.9386 ¢, ~49/40 = 351.0586 ¢, ~10/7 = 617.7065 ¢
error map: ⟨-0.061 +0.101 +0.096 -0.184 +0.075]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0806 ¢, ~10/7 = 617.7228 ¢
error map: ⟨0.000 +0.206 +0.213 -0.022 +0.269]

Optimal ET sequence: 41, 89, 130, 229, 270, 581, 670, 711, 981, 1251, 2232e

Badness (Sintel): 0.953

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4096/4095

Mapping: [⟨1 1 1 2 8 7], ⟨0 2 1 1 -12 -6], ⟨0 0 2 1 -2 -3]]

Optimal tunings:

  • WE: ~2 = 1199.9597 ¢, ~49/40 = 351.0718 ¢, ~10/7 = 617.6543 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0856 ¢, ~10/7 = 617.6715 ¢

Optimal ET sequence: 41, 89, 99, 130, 270, 581, 711, 981, 1292, 1562

Badness (Sintel): 0.542

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1729/1728, 2080/2079, 2401/2400

Subgroup-val mapping: [⟨1 1 1 2 8 7 0], ⟨0 2 1 1 -12 -6 11], ⟨0 0 2 1 -2 -3 2]]

Optimal tunings:

  • WE: ~2 = 1199.9752 ¢, ~49/40 = 351.0908 ¢, ~10/7 = 617.6473 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0984 ¢, ~10/7 = 617.6588 ¢

Optimal ET sequence: 41, 89, 130, 229, 270, 581, 851, 1562, 1832, 2413

Badness (Sintel): 0.488

Freyr

Freyr tempers out the symbiotic comma, so that the rastma is split in halves with each part standing in not only for 441/440~540/539, but the aberschisma. Freyr easily extends to the 13-limit, equating the major minthma with the aberschisma, and the no-17 19-limit, equating the undevicesimal schisma with it too.

It can be notated with neutral chain-of-fifths notation with two additional sets of accidentals, one for the generic comma step, and the other for the generic aberschisma step.

This temperament was named by Flora Canou in 2026, under the impression that it functions as a counterpart of freya.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 19712/19683

Mapping: [⟨1 1 1 2 -1], ⟨0 2 1 1 17], ⟨0 0 2 1 -1]]

Optimal tunings:

  • WE: ~2 = 1199.9474 ¢, ~49/40 = 351.1064 ¢, ~10/7 = 617.6483 ¢
error map: ⟨-0.053 +0.205 +0.037 -0.176 -0.104]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1156 ¢, ~10/7 = 617.6499 ¢
error map: ⟨0.000 +0.276 +0.102 -0.060 -0.003]

Optimal ET sequence: 41, 58, 99e, 171e, 212, 270, 581, 851, 1121, 1972, 2881bc

Badness (Sintel): 1.23

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 10648/10647

Mapping: [⟨1 1 1 2 -1 -2], ⟨0 2 1 1 17 23], ⟨0 0 2 1 -1 -2]]

Optimal tunings:

  • WE: ~2 = 1199.9432 ¢, ~49/40 = 351.1128 ¢, ~10/7 = 617.6419 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1232 ¢, ~10/7 = 617.6431 ¢

Optimal ET sequence: 41, 58, 99ef, 154cef, 171ef, 212, 270, 581, 851, 1490, 1760, 2071be, 2341b

Badness (Sintel): 0.760

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 2080/2079, 2401/2400

Mapping: [⟨1 1 1 2 -1 -2 0], ⟨0 2 1 1 17 23 11], ⟨0 0 2 1 -1 -2 2]]

Optimal tunings:

  • WE: ~2 = 1199.9435 ¢, ~49/40 = 351.1127 ¢, ~10/7 = 617.6404 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.1236 ¢, ~10/7 = 617.6459 ¢

Optimal ET sequence: 41, 58h, 99ef, 171ef, 212h, 270, 581, 851, 1490h, 1760h, 2071beh, 2341bh

Badness (Sintel): 0.523

Baldur

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800

Mapping: [⟨2 0 1 3 7], ⟨0 2 1 1 -2], ⟨0 0 2 1 3]]

Mapping generators: ~99/70, ~343/198, ~10/7

Optimal tunings:

  • WE: ~99/70 = 600.0149 ¢, ~343/198 = 950.9717 ¢, ~10/7 = 617.7007 ¢
error map: ⟨+0.030 -0.012 +0.074 -0.109 -0.055]
  • CWE: ~99/70 = 600.0000 ¢, ~343/198 = 950.9565 ¢, ~10/7 = 617.7017 ¢
error map: ⟨0.000 -0.042 +0.046 -0.168 -0.126]

Minimax tuning:

[[1 0 0 0 0⟩, [3/4 0 1/2 1/2 -1/2⟩, [0 0 1 0 0⟩, [23/16 0 5/8 1/8 -1/8⟩, [23/16 0 5/8 -7/8 7/8⟩]
unchanged-interval (eigenmonzo) basis: 2.5.11/7

Optimal ET sequence: 58, 72, 130, 198, 212, 270, 342, 612, 954, 1084, 1354, 1696, 4004de, 5700de

Badness (Sintel): 0.200

Projection pairs: 2 9801/4900, 3 117649/39204, 7 9801/1400, 11 913517247483640899/83082326424002500 to 5.7/2.99/4

Greenland

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715

Mapping: [⟨2 0 1 3 7 -1], ⟨0 2 1 1 -2 4], ⟨0 0 2 1 3 2]]

Optimal tunings:

  • WE: ~99/70 = 599.9868 ¢, ~26/15 = 951.0538 ¢, ~10/7 = 617.8062 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~26/15 = 951.0689 ¢, ~10/7 = 617.8067 ¢

Optimal ET sequence: 58, 72, 130, 198, 270, 940, 1210f

Badness (Sintel): 0.415

Projection pairs: 2 19600/9801, 3 676/225, 5 10400/2079, 7 20384000/2910897, 11 19208000000000000/1750211597736459, 13 5026736/385875 to 10/7.200/99.26/15

Midnatssol

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 442/441, 561/560, 676/675

Mapping: [⟨2 0 1 3 7 -1 5], ⟨0 2 1 1 -2 4 2], ⟨0 0 2 1 3 2 0]]

Optimal tunings:

  • WE: ~17/12 = 600.1118 ¢, ~26/15 = 951.1907 ¢, ~10/7 = 617.6100 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~26/15 = 951.0626 ¢, ~10/7 = 617.5842 ¢

Optimal ET sequence: 58, 72, 130, 140, 198, 212g, 270g, 342fg, 482fgg

Badness (Sintel): 0.720

Freya

Freya tempers out 3025/3024 and 41503/41472, so it splits breed's 28/27 into two 55/54~56/55's and 8/7 into two 77/72's. It also takes the 225/224~1029/1024 kleisma to represent 243/242, and splits it in half to get 385/384, 441/440, and 540/539. Since these are the commas tempered out by miracle, the simple relations between them make freya an elegant miracle detemperament.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024

Mapping: [⟨1 1 -1 1 5], ⟨0 2 3 2 1], ⟨0 0 4 2 -3]]

Mapping generators: ~2, ~49/40, ~84/55

Optimal tunings:

  • WE: ~2 = 1200.0491 ¢, ~49/40 = 350.9820 ¢, ~84/55 = 733.3514 ¢
error map: ⟨+0.030 -0.012 +0.074 -0.109 -0.055]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9890 ¢, ~84/55 = 733.3160 ¢
error map: ⟨0.000 -0.042 +0.046 -0.168 -0.126]

Minimax tuning:

Optimal ET sequence: 31, 41, 72, 167, 188, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee

Badness (Sintel): 0.204

Projection pairs: 3 2401/800, 5 22880495169/4575312500, 7 1058841/151250, 11 33275/3024 to 2.49/5.77/3

13-limit

Tridecimal freya maps 352/351 to the half-kleisma. It has miraculous as a sub-temperament.

Subgroup: 2.3.5.7.11.13

Comma list: 2401/2400, 3025/3024, 4096/4095

Mapping: [⟨1 1 -1 1 5 10], ⟨0 2 3 2 1 -9], ⟨0 0 4 2 -3 -6]]

Optimal tunings:

  • WE: ~2 = 1199.9870 ¢, ~49/40 = 351.0502 ¢, ~84/55 = 733.2969 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0534 ¢, ~84/55 = 733.3055 ¢

Optimal ET sequence: 31, 41, 72f, 198f, 229, 239, 270, 571, 581, 851, 882, 1152, 1463, 1733, 2615

Badness (Sintel): 0.800

Projection pairs: 3 2401/800, 5 22880495169/4575312500, 7 1058841/151250, 11 33275/3024, 13 1814078464000000000000000/139662717676432916098329 to 2.49/5.77/3

Eir

Eir maps 325/324, 364/363, and 729/728 to the half-kleisma and has manna as a sub-temperament. VIxen named this extension after a healer goddess or valkyrie from the Norse mythology, as it extends freya via the ibnsinma, which evokes associations with medicine.

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 3025/3024

Mapping: [⟨1 1 -1 1 5 5], ⟨0 2 3 2 1 6], ⟨0 0 4 2 -3 -5]]

Optimal tunings:

  • WE: ~2 = 1199.9994 ¢, ~49/40 = 351.0880 ¢, ~84/55 = 733.2478 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0880 ¢, ~84/55 = 733.2482 ¢

Optimal ET sequence: 41, 72, 157, 185cf, 198, 270, 581, 851, 1504, 1774f, 2085, 2355f

Badness (Sintel): 0.543

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1540/1539, 2080/2079, 2401/2400, 3025/3024

Subgroup-val mapping: [⟨1 1 -1 1 5 5 3], ⟨0 2 3 2 1 6 -2], ⟨0 0 4 2 -3 -5 3]]

Optimal tunings:

  • WE: ~2 = 1199.9951 ¢, ~49/40 = 351.0906 ¢, ~84/55 = 733.2475 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 351.0913 ¢, ~84/55 = 733.2473 ¢

Optimal ET sequence: 41, 72, 113, 157, 198, 270, 581, 851, 2085, 2355f

Badness (Sintel): 0.500

Heimlaug

Heimlaug maps 351/350 and 625/624 to the half-kleisma and has benediction as a subtemperament. VIxen named this extension after a völva (seeress) from the Gull-Þóris saga of Icelanders, as it extends freya via the fairytale comma, adding to the mystical theme.

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 3025/3024

Mapping: [⟨1 1 -1 1 5 -6], ⟨0 2 3 2 1 6], ⟨0 0 4 2 -3 13]]

Optimal tunings:

  • WE: ~2 = 1200.0601 ¢, ~49/40 = 350.9562 ¢, ~84/55 = 733.4471 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9646 ¢, ~84/55 = 733.4049 ¢

Optimal ET sequence: 31, 72, 103, 167, 198, 270, 571, 643, 913f

Badness (Sintel): 0.562

Vili

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 391314/390625

Mapping: [⟨1 1 -1 1 -4], ⟨0 2 3 2 6], ⟨0 0 6 3 14]]

Mapping generators: ~2, ~49/40, ~175/132

Optimal tunings:

  • WE: ~2 = 1199.9872 ¢, ~49/40 = 350.9740 ¢, ~175/132 = 488.9418 ¢
error map: ⟨-0.013 -0.020 +0.272 -0.065 -0.238]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9728 ¢, ~175/132 = 488.9475 ¢
error map: ⟨0.000 -0.009 +0.290 -0.038 -0.216]

Optimal ET sequence: 27e, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 670, 913, 1043, 1313, 1583

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 4225/4224

Mapping: [⟨1 1 -1 1 -4 3], ⟨0 2 3 2 6 1], ⟨0 0 6 3 14 1]]

Optimal tunings:

  • WE: ~2 = 1200.0597 ¢, ~49/40 = 350.9419 ¢, ~65/49 = 488.9741 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9441 ¢, ~65/49 = 488.9475 ¢

Optimal ET sequence: 27e, 37, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 913f

Badness (Sintel): 0.690

Frigg

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 644204/643125

Mapping: [⟨1 1 -7 -2 -7], ⟨0 2 3 2 4], ⟨0 0 10 5 11]]

Mapping generators: ~2, ~49/40, ~88/49

Optimal tunings:

  • WE: ~2 = 1200.0183 ¢, ~49/40 = 350.9854 ¢, ~88/49 = 1013.3709 ¢
error map: ⟨+0.018 +0.034 +0.223 -0.037 -0.425]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9871 ¢, ~88/49 = 1013.3556 ¢
error map: ⟨0.000 +0.019 +0.203 -0.074 -0.458]

Optimal ET sequence: 45e, 58, 103, 161, 212, 270, 643, 913, 1183e

Badness (Sintel): 2.15

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 10648/10647

Mapping: [⟨1 1 -7 -2 -7 -9], ⟨0 2 3 2 4 3], ⟨0 0 10 5 11 14]]

Optimal tunings:

  • WE: ~2 = 1200.0387 ¢, ~49/40 = 350.9502 ¢, ~70/39 = 1013.4065 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9526 ¢, ~70/39 = 1013.3747 ¢

Optimal ET sequence: 45ef, 58, 103, 161, 212, 270, 643, 913f, 1614ef *

* optimal patent val: 1241

Badness (Sintel): 0.873

Ennealimmic

Not to be confused with Ennealimnic.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374

Mapping: [⟨9 1 1 12 0], ⟨0 2 3 2 0], ⟨0 0 0 0 1]]

Mapping generators: ~27/25, ~5/3, ~11

Optimal tunings:

  • WE: ~27/25 = 133.3357 ¢, ~5/3 = 884.3288 ¢, ~11/8 = 551.2529 ¢
error map: ⟨+0.021 +0.038 +0.009 -0.139 -0.000]
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3215 ¢, ~11/8 = 551.2775 ¢
error map: ⟨0.000 +0.021 -0.016 -0.183 -0.040]

Optimal ET sequence: 72, 171(e), 198, 270, 342, 612, 954, 1323, 1395, 1665, 2007, 2277, 2619, 4284d, 6561dd, 6903dd

Badness (Sintel): 0.330

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4375/4374

Mapping: [⟨9 1 1 12 0 -31], ⟨0 2 3 2 0 5], ⟨0 0 0 0 1 1]]

Optimal tunings:

  • WE: ~27/25 = 133.3276 ¢, ~5/3 = 884.3660 ¢, ~11/8 = 551.7221 ¢
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3905 ¢, ~11/8 = 551.7085 ¢

Optimal ET sequence: 72, 171(ef), 198, 270, 639, 711, 981, 1692e

Badness (Sintel): 0.706

Retrieved from "https://en.xen.wiki/index.php?title=Breed_family&oldid=240102"