Very low accuracy temperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit

FloraC (talk | contribs)

4 user groups
27,346 edits
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,036 edits
Tags: Mobile edit Mobile web edit
Delete antonym, antony, and geryon, from the precedent of deletion of {7/6}. Perhaps {9/8} should be the threshold for tempering out superparticulars
 
(22 intermediate revisions by 4 users not shown)
Line 1: Line 1:
{{Technical data page}}
{{Technical data page}}
Below are listed some '''very low accuracy temperaments'''. Temperaments with exceedingly low accuracy such as these tend to [[tempering out|temper out]] very large intervals such as [[9/8]], [[10/9]], [[32/27]], or [[15/14]], equating wildly different interval sizes with [[semitone (interval size measure)|semitone]]-level or even greater [[error]]s. A lot of temperaments listed here don't have [[diamond monotone]] tunings because of their sheer inaccuracy. As a result, all of them are right on or even beyond the edge of what can be sensibly called a temperament at all; that is to say, they are [[exotemperament]]s.
Below are listed some '''very low accuracy temperaments'''. Temperaments with exceedingly low accuracy such as these tend to [[tempering out|temper out]] very large intervals such as [[9/8]], [[10/9]], [[32/27]], or [[15/14]], equating wildly different interval sizes with [[semitone (interval size measure)|semitone]]-level or even greater [[error]]s, and often swapping the sizes of simple ratios compared to just intonation. As a result, all of them are right on or even beyond the edge of what can be sensibly called a temperament at all; that is to say, they are [[exotemperament]]s.


== Antitonic ==
== Antitonic ==
Line 26: Line 26:
* [[:File:Antitonic(8).mp3|''Antitonic(8)'']] (2024) – short composition by [[Wensik]] in POTE-tuned 5-limit antitonic using an 8-note ternary scale.
* [[:File:Antitonic(8).mp3|''Antitonic(8)'']] (2024) – short composition by [[Wensik]] in POTE-tuned 5-limit antitonic using an 8-note ternary scale.


=== 7-limit ===
=== Septimal antitonic ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: 9/8, 15/14
Comma list: 9/8, 15/14


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


Optimal tunings:  
Optimal tunings:  
Line 46: Line 46:
Comma list: 9/8, 12/11, 15/14
Comma list: 9/8, 12/11, 15/14


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


Optimal tunings:  
Optimal tunings:  
Line 61: Line 61:
Comma list: 9/8, 11/10, 15/14
Comma list: 9/8, 11/10, 15/14


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


Optimal tunings:  
Optimal tunings:  
Line 76: Line 76:
Comma list: 9/8, 35/32
Comma list: 9/8, 35/32


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


Optimal tunings:  
Optimal tunings:  
Line 91: Line 91:
Comma list: 9/8, 12/11, 35/32
Comma list: 9/8, 12/11, 35/32


Mapping: {{mapping| 2 3 0 10 7 | 0 0 1 -1 0 }}
{{Mapping|legend=0| 2 3 0 10 7 | 0 0 1 -1 0 }}


Optimal tunings:  
Optimal tunings:  
Line 109: Line 109:


{{Mapping|legend=1| 3 5 0 | 0 0 1 }}
{{Mapping|legend=1| 3 5 0 | 0 0 1 }}
: mapping generators: ~4/3, ~5
: mapping generators: ~4/3, ~5


Line 122: Line 121:
[[Badness]] (Sintel): 1.83
[[Badness]] (Sintel): 1.83


== Yo (2c & 3) ==
== Antonian ==
This temperament family is characterized by the [[color notation|yo 2nd]] ([[10/9]]) being tempered out.
{{Main| Antonian }}
 
This temperament family is characterized by the [[color notation|yo 2nd]] ([[10/9]]) being tempered out. It identifies [[3/2]] with [[5/3]], [[4/3]] with [[6/5]], and [[5/4]] with [[9/8]].  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 140: Line 141:


[[Badness]] (Sintel): 0.392
[[Badness]] (Sintel): 0.392
=== Septimal antonian ===
{{See also| Trienstonic clan }}
Subgroup: 2.3.5.7
Comma list: 10/9, 15/14
{{Mapping|legend=0| 1 0 -1 -2 | 0 1 2 3 }}
Optimal tunings:
* WE: ~2 = 1193.691{{c}}, ~3/2 = 742.509{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.086{{c}}
{{Optimal ET sequence|legend=0| 2cd, 3d, 5c }}
Badness (Sintel): 0.606


=== Brutus ===
=== Brutus ===
Line 146: Line 164:
Comma list: 10/9, 28/25
Comma list: 10/9, 28/25


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


Optimal tunings:  
Optimal tunings:  
Line 157: Line 175:


=== Phlegyas ===
=== Phlegyas ===
{{See also| Archytas clan }}
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: 10/9, 35/32
Comma list: 10/9, 35/32


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


Optimal tunings:  
Optimal tunings:  
Line 172: Line 192:


=== Charon ===
=== Charon ===
{{See also| Jubilismic clan }}
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: 10/9, 49/45
Comma list: 10/9, 49/45


Mapping: {{mapping| 2 0 -2 -1 | 0 1 2 2 }}
{{Mapping|legend=0| 2 0 -2 -1 | 0 1 2 2 }}
: mapping generators: ~7/5, ~3
: mapping generators: ~7/5, ~3


Line 194: Line 216:
Comma list: 10/9, 49/48
Comma list: 10/9, 49/48


Mapping: {{mapping| 1 0 -1 2 | 0 2 4 1 }}
{{Mapping|legend=0| 1 0 -1 2 | 0 2 4 1 }}
: mapping generators: ~2, ~7/4
: mapping generators: ~2, ~7/4


Line 231: Line 253:
Comma list: 10/9, 16/15, 22/21
Comma list: 10/9, 16/15, 22/21


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


Optimal tunings:  
Optimal tunings:  
Line 242: Line 264:


== Quad ==
== Quad ==
Quad is identical to 4et in the 5-limit, but has an independent generator for prime 7.  
Quad is identical to [[4edo|4et]] in the 5-limit, but has an independent generator for prime 7.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 260: Line 282:
[[Badness]] (Sintel): 1.16
[[Badness]] (Sintel): 1.16


== Geryon ==
== Quint ==
{{See also| Dicot family }}
Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is already quite accurately approximated by 5edo for its size, this temperament provides little improvement, so it is only provided below for bookkeeping purposes. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.


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


[[Comma list]]: 8/7, 25/21
[[Comma list]]: 16/15, 27/25


{{Mapping|legend=1| 1 1 2 3 | 0 2 1 0 }}
{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
: Mapping generators: ~6/5, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1164.885{{c}}, ~5/4 = 374.162{{c}}
* [[WE]]: ~6/5 = 236.416{{c}}, ~7/4 = 1002.701{{c}} (~28/27 = 57.038{{c}})
: [[error map]]: {{val| -35.115 +11.253 -82.382 +125.830 }}
: [[error map]]: {{val| -17.921 -10.628 +50.676 -1.966 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 375.277{{c}}
* [[CWE]]: ~6/5 = 240.000{{c}}, ~7/4 = 1005.135{{c}} (~28/27 = 45.135{{c}})
: error map: {{val| 0.000 +48.600 -11.036 +231.174 }}
: error map: {{val| 0.000 +18.045 +93.686 +36.309 }}


{{Optimal ET sequence|legend=1| 1b, 2b, 3d }}
{{Optimal ET sequence|legend=1| 5, 15ccd, 20cccd }}


[[Badness]] (Sintel): 1.29
[[Badness]] (Sintel): 1.22


== Malacoda ==
== Malacoda ==
Line 287: Line 310:


{{Mapping|legend=1| 1 0 3 2 | 0 2 -1 1 }}
{{Mapping|legend=1| 1 0 3 2 | 0 2 -1 1 }}
: mapping generators: ~2, ~7/4
: mapping generators: ~2, ~7/4


Line 343: Line 365:
Comma list: 15/14, 22/21, 33/32
Comma list: 15/14, 22/21, 33/32


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


Optimal tunings:  
Optimal tunings:  
Line 373: Line 395:


== Plutus ==
== Plutus ==
{{Distinguish| Pluto }}
{{See also| Meantone family }}
{{See also| Meantone family }}


Line 396: Line 419:
Comma list: 15/14, 22/21, 81/80
Comma list: 15/14, 22/21, 81/80


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


Optimal tunings:  
Optimal tunings:  
Line 405: Line 428:


Badness (Sintel): 1.08
Badness (Sintel): 1.08
== Involution ==
Involution tempers out the [[45/32|ptolemaic augmented fourth (45/32)]]. Its ploidacot is monocot. It was named by [[User:CompactStar|CompactStar]] in 2023.
[[Subgroup]]: 2.3.5
[[Comma list]]: [[45/32]]
{{Mapping|legend=1| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1265.406{{c}}, ~3/2 = 552.346{{c}}
: [[error map]]: {{val| +65.405 -84.203 -94.789 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 520.626{{c}}
: error map: {{val| 0.000 -181.329 -227.565 }}
{{Optimal ET sequence|legend=1| 2c, 5bc, 7bbcc }}
[[Badness]] (Sintel): 3.12
=== 7-limit ===
Subgroup: 2.3.5.7
Comma list: 8/7, 45/28
Mapping: {{mapping| 1 0 5 3 | 0 1 -2 0 }}
Optimal tunings:
* WE: ~2 = 1205.230{{c}}, ~3/2 = 517.557{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 515.099{{c}}
Optimal ET sequence: none
Badness (Sintel): 3.01
== Devolution ==
Devolution was named by [[User:Akselai|Akselai]] in 2024.
[[Subgroup]]: 3.5.7
[[Comma list]]: [[35/27]]
{{Mapping|legend=1| 1 0 3 | 0 1 -1 }}
: mapping generators: ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1988.549{{c}}, ~5/3 = 719.679{{c}}
: [[error map]]: {{val| +86.594 -78.085 -111.407 }}
* [[CWE]]: ~3 = 1901.955{{c}}, ~5/3 = 691.468{{c}}
: error map: {{val| 0.000 -192.891 -256.384 }}
[[Optimal ET sequence]]: [[1edt|b1]], [[2edt|b2d]], [[3edt|b3]], [[8edt|b8cdd]]
[[Badness]] (Sintel): 0.751


== Codex ==
== Codex ==
{{See also| Bug family }}
{{See also| Bug family }}
Codex was named by [[User:Jerdle|Jerdle]] as an abbreviation of "co-decimal", as it resembles [[decimal]] in many ways, but exchanges the roles of 5's and 7's. While decimal equates [[8/7]] and [[7/6]], as well as [[6/5]] and [[5/4]], this equates [[10/9]] and [[6/5]], as well as [[7/6]] and [[9/7]]. It is an extension of [[bug]] and [[54/49|mujannabic]] in the same way decimal is of [[semaphore]] and [[dicot]]. Its ploidacot is diploid beta-dicot.  
Codex was named by [[User:Jerdle|Jerdle]] as an abbreviation of "co-decimal", as it resembles [[decimal]] in many ways, but exchanges the roles of 5's and 7's. While decimal equates [[8/7]] and [[7/6]], as well as [[6/5]] and [[5/4]], this equates [[10/9]] and [[6/5]], as well as [[7/6]] and [[9/7]]. It is an extension of [[bug]] and [[54/49|mujannabic]] in the same way decimal is of [[semaphore]] and [[dicot]]. Its ploidacot is diploid dicot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 485: Line 451:
== Oxygen ==
== Oxygen ==
{{See also| Porcupine family }}
{{See also| Porcupine family }}
Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7. While this means it does not represent either of those intervals with any real accuracy, it is still of interest because its comma basis suggests potential utility to construct [[fokker block|Fokker blocks]].
Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7, and conflating 10/9 with 7/6. While this means it does not represent 7-limit JI with any real accuracy, it is still of interest because its comma basis suggests potential utility to construct [[fokker block|Fokker blocks]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 499: Line 465:
: error map: {{val| 0.000 -0.083 -16.526 -100.911 }}
: error map: {{val| 0.000 -0.083 -16.526 -100.911 }}


{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}
{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}*
 
<nowiki/>* [[optimal patent val]]: [[8edo|8]]


[[Badness]] (Sintel): 1.52
[[Badness]] (Sintel): 1.52
== Bixby ==
{{Main| Bixby }}
[[Subgroup]]: 2.3.5
[[Comma list]]: [[4/3]]
{{Mapping|legend=1| 1 2 0 | 0 0 1 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1020.058{{c}}, ~5/4 = 674.394{{c}}
: [[error map]]: {{val| -179.942 +138.161 -71.803 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 629.521{{c}}
: error map: {{val| 0.000 +498.045 +243.208 }}
{{Optimal ET sequence|legend=1| 1c, 2b, 3bbcc }}
[[Badness]] (Sintel): 0.424
== Archon ==
{{Main| Archon }}
[[Subgroup]]: 2.3.5
[[Comma list]]: [[5/4]]
{{Mapping|legend=1| 1 0 2 | 0 1 0 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1268.274{{c}}, ~3/2 = 612.921{{c}}
: [[error map]]: {{val| +68.274 -20.760 -249.765 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 614.055{{c}}
: error map: {{val| 0.000 -87.900 -386.314 }}
{{Optimal ET sequence|legend=1| 2c }}
[[Badness]] (Sintel): 0.474
== Seesaw ==
{{Main| Seesaw }}
Seesaw tempers out the [[6/5|classic minor third (6/5)]], equating the [[5/1|fifth]] and [[6/1|sixth harmonic]]s. It was named by [[User:Xenllium|Xenllium]] in 2026.
[[Subgroup]]: 2.3.5
[[Comma list]]: 6/5
{{Mapping|legend=1| 1 0 1 | 0 1 1 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1155.569{{c}}, ~3/2 = 643.349{{c}}
: [[error map]]: {{val| -44.431 -103.037 +168.173 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 627.511{{c}}
: error map: {{val| 0.000 -74.444 +241.197 }}
{{Optimal ET sequence|legend=1| 2 }}
[[Badness]] (Sintel): 0.367
=== 2.3.5.11 subgroup ===
This temperament is extended to the 2.3.5.11 subgroup naturally, tempering out both [[11/10]] and [[12/11]], undecimal neutral seconds.
Subgroup: 2.3.5.11
Comma list: 6/5, 11/10
Mapping: {{mapping| 1 0 1 2 | 0 1 1 1 }}
Optimal tunings:
* WE: ~2 = 1156.418{{c}}, ~3/2 = 643.202{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 627.023{{c}}
{{Optimal ET sequence|legend=0| 2 }}
Badness (Sintel): 0.499
=== Heavy windmill ===
Heavy windmill tempers out [[9/7]] and [[15/14]] in the 7-limit.
Subgroup: 2.3.5.7
Comma list: 6/5, 9/7
Mapping: {{mapping| 1 0 1 0 | 0 1 1 2 }}
Optimal tunings:
* WE: ~2 = 1161.600{{c}}, ~3/2 = 571.169{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 559.563{{c}}
{{Optimal ET sequence|legend=0| 2 }}
Badness (Sintel): 0.676
==== 11-limit ====
Subgroup: 2.3.5.7.11
Comma list: 6/5, 9/7, 11/10
Mapping: {{mapping| 1 0 1 0 2 | 0 1 1 2 1 }}
Optimal tunings:
* WE: ~2 = 1166.584{{c}}, ~3/2 = 568.073{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 558.941{{c}}
{{Optimal ET sequence|legend=0| 2 }}
Badness (Sintel): 0.774
=== Light windmill ===
Light windmill tempers out [[8/7]] and [[21/20]] in the 7-limit.
Subgroup: 2.3.5.7
Comma list: 6/5, 8/7
Mapping: {{mapping| 1 0 1 3 | 0 1 1 0 }}
Optimal tunings:
* WE: ~2 = 1134.018{{c}}, ~3/2 = 670.285{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 667.893{{c}}
{{Optimal ET sequence|legend=0| 2 }}
Badness (Sintel): 0.629
==== 11-limit ====
Subgroup: 2.3.5.7.11
Comma list: 6/5, 8/7, 11/10
Mapping: {{mapping| 1 0 1 3 2 | 0 1 1 0 1 }}
Optimal tunings:
* WE: ~2 = 1136.109{{c}}, ~3/2 = 672.403{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 668.374{{c}}
{{Optimal ET sequence|legend=0| 2 }}
Badness (Sintel): 0.681
[[Category:Temperament collections]]

Latest revision as of 09:30, 20 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.

Below are listed some very low accuracy temperaments. Temperaments with exceedingly low accuracy such as these tend to temper out very large intervals such as 9/8, 10/9, 32/27, or 15/14, equating wildly different interval sizes with semitone-level or even greater errors, and often swapping the sizes of simple ratios compared to just intonation. As a result, all of them are right on or even beyond the edge of what can be sensibly called a temperament at all; that is to say, they are exotemperaments.

Antitonic

This temperament is characterized by 9/8 being tempered out and has been termed a "troll temperament" by its namers. Its ploidacot is diploid acot. It is named on account of 4/3 and 3/2 both being represented by the 600-cent half octave, which, in terms of diatonic function, serves as an antitonic. The 7-limit extension tempers out 15/14 and 21/20, equating 5/4 with 7/6 and 6/5 with 8/7. The 11-limit extension tempers out 12/11 and 33/32. The original 5-limit is basically the 3-limit music of 2edo with the addition of harmonic 5 represented by an independent generator.

Subgroup: 2.3.5

Comma list: 9/8

Mapping: [⟨2 3 0], ⟨0 0 1]]

mapping generators: ~3/2, ~5

Optimal tunings:

  • WE: ~3/2 = 615.125 ¢, ~5/4 = 321.597 ¢ (~6/5 = 293.528 ¢)
error map: ⟨+30.250 -56.581 -4.217]
  • CWE: ~3/2 = 600.000 ¢, ~5/4 = 336.527 ¢ (~6/5 = 263.473 ¢)
error map: ⟨0.000 -101.955 -49.787]

Optimal ET sequence: 2, 4

Badness (Sintel): 0.508

Music
  • Antitonic(8) (2024) – short composition by Wensik in POTE-tuned 5-limit antitonic using an 8-note ternary scale.

Septimal antitonic

Subgroup: 2.3.5.7

Comma list: 9/8, 15/14

Mapping: [⟨2 3 0 1], ⟨0 0 1 1]]

Optimal tunings:

  • WE: ~3/2 = 614.759 ¢, ~7/6 = 309.426 ¢ (~6/5 = 305.334 ¢)
  • CWE: ~3/2 = 600.000 ¢, ~7/6 = 326.047 ¢ (~6/5 = 273.953 ¢)

Optimal ET sequence: 2, 4

Badness (Sintel): 0.490

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9/8, 12/11, 15/14

Mapping: [⟨2 3 0 1 7], ⟨0 0 1 1 0]]

Optimal tunings:

  • WE: ~3/2 = 606.293 ¢, ~5/4 = 343.862 ¢ (~8/7 = 262.431 ¢)
  • CWE: ~3/2 = 600.000 ¢, ~5/4 = 348.102 ¢ (~8/7 = 251.898 ¢)

Optimal ET sequence: 2, 4

Badness (Sintel): 0.721

Antietam

Subgroup: 2.3.5.7.11

Comma list: 9/8, 11/10, 15/14

Mapping: [⟨2 3 0 1 2], ⟨0 0 1 1 1]]

Optimal tunings:

  • WE: ~3/2 = 616.135 ¢, ~5/4 = 330.741 ¢ (~6/5 = 285.393 ¢)
  • CWE: ~3/2 = 600.000 ¢, ~5/4 = 349.843 ¢ (~8/7 = 250.157 ¢)

Optimal ET sequence: 2, 4e

Badness (Sintel): 0.628

Antaeus

Subgroup: 2.3.5.7

Comma list: 9/8, 35/32

Mapping: [⟨2 3 0 10], ⟨0 0 1 -1]]

Optimal tunings:

  • WE: ~3/2 = 614.854 ¢, ~5/4 = 323.784 ¢ (~6/5 = 291.070 ¢)
  • CWE: ~3/2 = 600.000 ¢, ~5/4 = 317.349 ¢ (~6/5 = 282.651 ¢)

Optimal ET sequence: 2d, 4

Badness (Sintel): 0.950

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9/8, 12/11, 35/32

Mapping: [⟨2 3 0 10 7], ⟨0 0 1 -1 0]]

Optimal tunings:

  • WE: ~3/2 = 609.311 ¢, ~5/4 = 323.110 ¢ (~6/5 = 286.200 ¢)
  • CWE: ~3/2 = 600.000 ¢, ~5/4 = 318.904 ¢ (~6/5 = 281.096 ¢)

Optimal ET sequence: 2d, 4

Badness (Sintel): 1.12

Alteraugment

Alteraugment tempers out the Pythagorean minor third (32/27). It is like augmented, but the period represents 4/3 instead of 5/4, and the generator in turn provides 5/4 instead of 3/2. Its ploidacot is triploid acot. Vector Graphics suggests the name kinsborough for this temperament.

Subgroup: 2.3.5

Comma list: 32/27

Mapping: [⟨3 5 0], ⟨0 0 1]]

mapping generators: ~4/3, ~5

Optimal tunings:

  • WE: ~4/3 = 389.212 ¢, ~5/4 = 447.530 ¢ (~10/9 = 58.318 ¢)
error map: ⟨-32.364 +44.105 -3.512]
  • CWE: ~4/3 = 400.000 ¢, ~5/4 = 434.191 ¢ (~15/16 = 34.191 ¢)
error map: ⟨0.000 +98.045 +47.878]

Optimal ET sequence: 3, 12bcc, 15bbcc

Badness (Sintel): 1.83

Antonian

Main article: Antonian

This temperament family is characterized by the yo 2nd (10/9) being tempered out. It identifies 3/2 with 5/3, 4/3 with 6/5, and 5/4 with 9/8.

Subgroup: 2.3.5

Comma list: 10/9

Mapping: [⟨1 0 -1], ⟨0 1 2]]

Optimal tunings:

  • WE: ~2 = 1187.236 ¢, ~3/2 = 767.716 ¢
error map: ⟨-12.764 +52.997 -63.645]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 767.718 ¢
error map: ⟨0.000 +65.763 -50.877]

Optimal ET sequence: 2c, 3

Badness (Sintel): 0.392

Septimal antonian

See also: Trienstonic clan

Subgroup: 2.3.5.7

Comma list: 10/9, 15/14

Mapping: [⟨1 0 -1 -2], ⟨0 1 2 3]]

Optimal tunings:

  • WE: ~2 = 1193.691 ¢, ~3/2 = 742.509 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 743.086 ¢

Optimal ET sequence: 2cd, 3d, 5c

Badness (Sintel): 0.606

Brutus

Subgroup: 2.3.5.7

Comma list: 10/9, 28/25

Mapping: [⟨1 0 -1 -4], ⟨0 1 2 4]]

Optimal tunings:

  • WE: ~2 = 1158.982 ¢, ~3/2 = 819.228 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 831.346 ¢

Optimal ET sequence: 3, 7bc

Badness (Sintel): 1.35

Phlegyas

See also: Archytas clan

Subgroup: 2.3.5.7

Comma list: 10/9, 35/32

Mapping: [⟨1 0 -1 6], ⟨0 1 2 -2]]

Optimal tunings:

  • WE: ~2 = 1206.510 ¢, ~3/2 = 747.166 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 743.797 ¢

Optimal ET sequence: 3, 5c, 8c

Badness (Sintel): 1.30

Charon

See also: Jubilismic clan

Subgroup: 2.3.5.7

Comma list: 10/9, 49/45

Mapping: [⟨2 0 -2 -1], ⟨0 1 2 2]]

mapping generators: ~7/5, ~3

Optimal tunings:

  • WE: ~7/5 = 593.832 ¢, ~3/2 = 774.559 ¢ (~15/14 = 180.726 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~3/2 = 774.466 ¢ (~15/14 = 174.466 ¢)

Optimal ET sequence: 4bcd, 6

Badness (Sintel): 1.43

Nessus

See also: Semaphoresmic clan

Subgroup: 2.3.5.7

Comma list: 10/9, 49/48

Mapping: [⟨1 0 -1 2], ⟨0 2 4 1]]

mapping generators: ~2, ~7/4

Optimal tunings:

  • WE: ~2 = 1189.201 ¢, ~7/4 = 978.002 ¢
  • CWE: ~2 = 1200.000 ¢, ~7/4 = 983.918 ¢

Optimal ET sequence: 5c, 6

Badness (Sintel): 1.49

Ternary

Ternary is identical to 3et in the 5-limit, but has an independent generator for prime 7.

Subgroup: 2.3.5.7

Comma list: 10/9, 16/15

Mapping: [⟨3 5 7 0], ⟨0 0 0 1]]

mapping generators: ~5/4, ~7

Optimal tunings:

  • WE: ~5/4 = 391.796 ¢, ~7/4 = 1012.806 ¢ (~15/14 = 162.582 ¢)
error map: ⟨-24.612 +57.026 -43.741 -5.243]
  • CWE: ~5/4 = 400.000 ¢, ~7/4 = 1016.378 ¢ (~8/7 = 183.622 ¢)
error map: ⟨0.000 +98.045 +13.686 +47.552]

Optimal ET sequence: 3d, 6, 9bd

Badness (Sintel): 0.726

11-limit

Subgroup: 2.3.5.7.11

Comma list: 10/9, 16/15, 22/21

Mapping: [⟨3 5 7 0 2], ⟨0 0 0 1 1]]

Optimal tunings:

  • WE: ~5/4 = 391.788 ¢, ~7/4 = 1011.942 ¢ (~12/11 = 163.422 ¢)
  • CWE: ~5/4 = 400.000 ¢, ~7/4 = 1013.973 ¢ (~12/11 = 186.027 ¢)

Optimal ET sequence: 3de, 6

Badness (Sintel): 0.846

Quad

Quad is identical to 4et in the 5-limit, but has an independent generator for prime 7.

Subgroup: 2.3.5.7

Comma list: 9/8, 25/24

Mapping: [⟨4 6 9 0], ⟨0 0 0 1]]

Optimal tunings:

  • WE: ~6/5 = 308.074 ¢, ~7/4 = 963.670 ¢
error map: ⟨+32.295 -53.513 -13.650 -5.150]
  • CWE: ~6/5 = 300.000 ¢, ~7/4 = 897.589 ¢
error map: ⟨0.000 -101.955 -86.314 -71.236]

Optimal ET sequence: 4

Badness (Sintel): 1.16

Quint

Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is already quite accurately approximated by 5edo for its size, this temperament provides little improvement, so it is only provided below for bookkeeping purposes. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what blackwood does.

Subgroup: 2.3.5.7

Comma list: 16/15, 27/25

Mapping: [⟨5 8 12 0], ⟨0 0 0 1]]

Mapping generators: ~6/5, ~7

Optimal tunings:

  • WE: ~6/5 = 236.416 ¢, ~7/4 = 1002.701 ¢ (~28/27 = 57.038 ¢)
error map: ⟨-17.921 -10.628 +50.676 -1.966]
  • CWE: ~6/5 = 240.000 ¢, ~7/4 = 1005.135 ¢ (~28/27 = 45.135 ¢)
error map: ⟨0.000 +18.045 +93.686 +36.309]

Optimal ET sequence: 5, 15ccd, 20cccd

Badness (Sintel): 1.22

Malacoda

See also: Semaphoresmic clan

Subgroup: 2.3.5.7

Comma list: 15/14, 35/32

Mapping: [⟨1 0 3 2], ⟨0 2 -1 1]]

mapping generators: ~2, ~7/4

Optimal tunings:

  • WE: ~2 = 1223.542 ¢, ~7/4 = 941.838 ¢
error map: ⟨+23.542 -18.278 -57.528 +20.096]
  • CWE: ~2 = 1200.000 ¢, ~7/4 = 927.096 ¢
error map: ⟨0.000 -47.763 -113.410 -41.730]

Optimal ET sequence: 1, 3b, 4, 9c, 13bcc

Badness (Sintel): 0.942

Ugolino

See also: Bug family

Subgroup: 2.3.5.7

Comma list: 15/14, 27/25

Mapping: [⟨1 0 0 -1], ⟨0 2 3 5]]

Optimal tunings:

  • WE: ~2 = 1206.628 ¢, ~7/4 = 926.730 ¢
error map: ⟨+6.628 -48.494 -6.122 +58.198]
  • CWE: ~2 = 1200.000 ¢, ~7/4 = 923.776 ¢
error map: ⟨0.000 -54.403 -14.986 +50.054]

Optimal ET sequence: 1cdd, 3bcdd, 4, 9d

Badness (Sintel): 1.11

Medusa

See also: Archytas clan and Mavila family

Subgroup: 2.3.5.7

Comma list: 15/14, 64/63

Mapping: [⟨1 0 7 6], ⟨0 1 -3 -2]]

Optimal tunings:

  • WE: ~2 = 1200.960 ¢, ~3/2 = 686.181 ¢
error map: ⟨+0.960 -14.814 -41.014 +62.655]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 685.511 ¢
error map: ⟨0.000 -16.443 -42.849 +60.150]

Optimal ET sequence: 2, 5c, 7

Badness (Sintel): 1.08

11-limit

Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 33/32

Mapping: [⟨1 0 7 6 5], ⟨0 1 -3 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1202.757 ¢, ~3/2 = 687.384 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 685.462 ¢

Optimal ET sequence: 2, 5c, 7

Badness (Sintel): 0.887

Wallaby

See also: Trienstonic clan and Mavila family

Subgroup: 2.3.5.7

Comma list: 28/27, 35/32

Mapping: [⟨1 0 7 -2], ⟨0 1 -3 3]]

Optimal tunings:

  • WE: ~2 = 1216.024 ¢, ~3/2 = 700.583 ¢
error map: ⟨+16.024 +14.652 -23.967 -51.053]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 691.757 ¢
error map: ⟨0.000 -10.198 -61.585 -93.555]

Optimal ET sequence: 2d, 5c, 7d, 19ccdd

Badness (Sintel): 1.48

Plutus

Not to be confused with Pluto.
See also: Meantone family

Subgroup: 2.3.5.7

Comma list: 15/14, 81/80

Mapping: [⟨1 0 -4 -5], ⟨0 1 4 5]]

Optimal tunings:

  • WE: ~2 = 1203.936 ¢, ~3/2 = 685.135 ¢
error map: ⟨+3.936 -12.884 -45.774 +56.849]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 683.935 ¢
error map: ⟨0.000 -18.020 -50.573 +50.850]

Optimal ET sequence: 2cd, 5d, 7

Badness (Sintel): 1.14

11-limit

Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 81/80

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

Optimal tunings:

  • WE: ~2 = 1203.293 ¢, ~3/2 = 687.114 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 686.078 ¢

Optimal ET sequence: 2cde, 5de, 7

Badness (Sintel): 1.08

Codex

See also: Bug family

Codex was named by Jerdle as an abbreviation of "co-decimal", as it resembles decimal in many ways, but exchanges the roles of 5's and 7's. While decimal equates 8/7 and 7/6, as well as 6/5 and 5/4, this equates 10/9 and 6/5, as well as 7/6 and 9/7. It is an extension of bug and mujannabic in the same way decimal is of semaphore and dicot. Its ploidacot is diploid dicot.

Subgroup: 2.3.5.7

Comma list: 27/25, 50/49

Mapping: [⟨2 0 0 1], ⟨0 2 3 3]]

Optimal tunings:

  • WE: ~7/5 = 598.589 ¢, ~5/3 = 934.978 ¢
error map: ⟨-2.821 -31.998 +18.621 +34.699]
  • CWE: ~7/5 = 600.000 ¢, ~5/3 = 936.030 ¢
error map: ⟨0.000 -29.895 +21.776 +39.264]

Optimal ET sequence: 4, 10cd, 14d

Badness (Sintel): 1.95

Oxygen

See also: Porcupine family

Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7, and conflating 10/9 with 7/6. While this means it does not represent 7-limit JI with any real accuracy, it is still of interest because its comma basis suggests potential utility to construct Fokker blocks.

Subgroup: 2.3.5.7

Comma list: 21/20, 175/162

Mapping: [⟨1 2 3 3], ⟨0 -3 -5 -2]]

Optimal tunings:

  • WE: ~2 = 1213.695 ¢, ~10/9 = 171.042 ¢
error map: ⟨+13.695 +12.309 -0.438 -69.825]
  • CWE: ~2 = 1200.000 ¢, ~10/9 = 166.042 ¢
error map: ⟨0.000 -0.083 -16.526 -100.911]

Optimal ET sequence: 1c, …, 6bcd, 7d*

* optimal patent val: 8

Badness (Sintel): 1.52

Retrieved from "https://en.xen.wiki/index.php?title=Very_low_accuracy_temperaments&oldid=240144"