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Ragismic microtemperaments: Difference between revisions

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If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example). In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS.
If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example). In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS.
Subgroup: 2.3.5.7


[[Comma list]]: 2401/2400, 4375/4374
[[Comma list]]: 2401/2400, 4375/4374
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=== 11-limit ===
=== 11-limit ===
The ennealimmal temperament can be described as 99e&270 temperament, which tempers out 5632/5625 (vishdel comma) and 19712/19683 (symbiotic comma).
The ennealimmal temperament can be described as 99e&270 temperament, which tempers out 5632/5625 (vishdel comma) and 19712/19683 (symbiotic comma).
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 4375/4374, 5632/5625
Comma list: 2401/2400, 4375/4374, 5632/5625
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==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374
Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374


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=== Ennealimmia ===
=== Ennealimmia ===
Ennealimmal temperament has various extensions to the 11-limit. Tempering out 131072/130977 (salururu comma) leads to the ''ennealimmia'' temperament (171&270, named by [[User:Xenllium|Xenllium]]).
Ennealimmal temperament has various extensions to the 11-limit. Tempering out 131072/130977 (salururu comma) leads to the ''ennealimmia'' temperament (171&270, named by [[User:Xenllium|Xenllium]]).
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 4375/4374, 131072/130977
Comma list: 2401/2400, 4375/4374, 131072/130977
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==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374
Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374


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=== Ennealimnic ===
=== Ennealimnic ===
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 4375/4356
Comma list: 243/242, 441/440, 4375/4356


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==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 364/363, 441/440, 625/624
Comma list: 243/242, 364/363, 441/440, 625/624


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===== 17-limit =====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 364/363, 375/374, 441/440, 595/594
Comma list: 243/242, 364/363, 375/374, 441/440, 595/594


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==== Ennealim ====
==== Ennealim ====
Subgroup: 2.3.5.7.13
Comma list: 169/168, 243/242, 325/324, 441/440
Comma list: 169/168, 243/242, 325/324, 441/440


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=== Ennealiminal ===
=== Ennealiminal ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 4375/4374
Comma list: 385/384, 1375/1372, 4375/4374


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==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 325/324, 385/384, 1375/1372
Comma list: 169/168, 325/324, 385/384, 1375/1372


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=== Hemiennealimmal ===
=== Hemiennealimmal ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 4375/4374
Comma list: 2401/2400, 3025/3024, 4375/4374


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==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024
Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024


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==== Semihemiennealimmal ====
==== Semihemiennealimmal ====
Subgroup: 2.3.5.7.11.13
Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374
Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374


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=== Semiennealimmal ===
=== Semiennealimmal ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 4000/3993, 4375/4374
Comma list: 2401/2400, 4000/3993, 4375/4374


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==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374
Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374


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=== Quadraennealimmal ===
=== Quadraennealimmal ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 4375/4374, 234375/234256
Comma list: 2401/2400, 4375/4374, 234375/234256


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=== Trinealimmal ===
=== Trinealimmal ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 4375/4374, 2097152/2096325
Comma list: 2401/2400, 4375/4374, 2097152/2096325


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== Gamera ==
== Gamera ==
Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, 589824/588245
[[Comma list]]: 4375/4374, 589824/588245


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=== Hemigamera ===
=== Hemigamera ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 589824/588245
Comma list: 3025/3024, 4375/4374, 589824/588245


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==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024


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== Supermajor ==
== Supermajor ==
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of {{multival|37 46 75 -13 15 45}}. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of {{multival|37 46 75 -13 15 45}}. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 52734375/52706752
[[Comma list]]: 4375/4374, 52734375/52706752
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=== Semisupermajor ===
=== Semisupermajor ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 35156250/35153041
Comma list: 3025/3024, 4375/4374, 35156250/35153041


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== Enneadecal ==
== Enneadecal ==
Enneadecal temperament tempers out the enneadeca, {{monzo|-14 -19 19}}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo|19EDO]] up to just ones. [[171edo|171EDO]] is a good tuning for either the 5 or 7 limits, and [[494edo|494EDO]] shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo|665EDO]] for a tuning.
Enneadecal temperament tempers out the enneadeca, {{monzo|-14 -19 19}}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo|19EDO]] up to just ones. [[171edo|171EDO]] is a good tuning for either the 5 or 7 limits, and [[494edo|494EDO]] shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo|665EDO]] for a tuning.
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 703125/702464
[[Comma list]]: 4375/4374, 703125/702464
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=== Hemienneadecal ===
=== Hemienneadecal ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 234375/234256
Comma list: 3025/3024, 4375/4374, 234375/234256


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==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213


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== Deca ==
== Deca ==
Deca temperament has a period of 1/10 octave and tempers out the [[15/14ths equal temperament #Linus temperaments|linus comma]], {{monzo|11 -10 -10 10}} and {{monzo|12 -3 -14 9}} = 165288374272/164794921875 (satritrizo-asepbigu).
Deca temperament has a period of 1/10 octave and tempers out the [[15/14ths equal temperament #Linus temperaments|linus comma]], {{monzo|11 -10 -10 10}} and {{monzo|12 -3 -14 9}} = 165288374272/164794921875 (satritrizo-asepbigu).
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 165288374272/164794921875
[[Comma list]]: 4375/4374, 165288374272/164794921875
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=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 422576/421875
Comma list: 3025/3024, 4375/4374, 422576/421875


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=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374


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== Mitonic ==
== Mitonic ==
{{see also|Minortonic family #Mitonic}}
{{see also|Minortonic family #Mitonic}}
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 2100875/2097152
[[Comma list]]: 4375/4374, 2100875/2097152
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== Sfourth ==
== Sfourth ==
Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, 64827/64000
[[Comma list]]: 4375/4374, 64827/64000


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=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 121/120, 441/440, 4375/4374
Comma list: 121/120, 441/440, 4375/4374


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==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 169/168, 325/324, 441/440
Comma list: 121/120, 169/168, 325/324, 441/440


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=== Sfour ===
=== Sfour ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 2401/2376, 4375/4374
Comma list: 385/384, 2401/2376, 4375/4374


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==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 364/363, 385/384, 4375/4374
Comma list: 196/195, 364/363, 385/384, 4375/4374


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== Abigail ==
== Abigail ==
Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, 2147483648/2144153025
[[Comma list]]: 4375/4374, 2147483648/2144153025


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=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 20614528/20588575
Comma list: 3025/3024, 4375/4374, 20614528/20588575


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=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095


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== Semidimi ==
== Semidimi ==
The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo|-12 -73 55}} and 7-limit 3955078125/3954653486, as well as 4375/4374.
The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo|-12 -73 55}} and 7-limit 3955078125/3954653486, as well as 4375/4374.
Subgroup: 2.3.5


[[Comma]]: {{monzo|-12 -73 55}}
[[Comma]]: {{monzo|-12 -73 55}}
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=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, 3955078125/3954653486
[[Comma list]]: 4375/4374, 3955078125/3954653486


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== Brahmagupta ==
== Brahmagupta ==
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo|47 -7 -7 -7}} = 140737488355328 / 140710042265625.
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo|47 -7 -7 -7}} = 140737488355328 / 140710042265625.
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 70368744177664/70338939985125
[[Comma list]]: 4375/4374, 70368744177664/70338939985125
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=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 4000/3993, 4375/4374, 131072/130977
Comma list: 4000/3993, 4375/4374, 131072/130977


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=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374


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== Quasithird ==
== Quasithird ==
The '''quasithird''' temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}.
The '''quasithird''' temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}.
Subgroup: 2.3.5


[[Comma]]: {{monzo|55 -64 20}}
[[Comma]]: {{monzo|55 -64 20}}
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[[POTE generator]]: ~1594323/1280000 = 380.395
[[POTE generator]]: ~1594323/1280000 = 380.395


[[Vals]]: {{Val list| 60, 164, 224, 388, 612, 836, 1000, 1448, 1612, 2224, 2836 }}
{{Val list|legend=1| 60, 164, 224, 388, 612, 836, 1000, 1448, 1612, 2224, 2836 }}


[[Badness]]: 0.099519
[[Badness]]: 0.099519


=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, 1153470752371588581/1152921504606846976
[[Comma list]]: 4375/4374, 1153470752371588581/1152921504606846976


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[[POTE generator]]: ~5103/4096 = 380.388  
[[POTE generator]]: ~5103/4096 = 380.388  


[[Vals]]: {{Val list| 60d, 164, 224, 388, 612, 1448, 2060 }}
{{Val list|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}


[[Badness]]: 0.061813
[[Badness]]: 0.061813


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296


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=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 2200/2197, 3025/3024, 4375/4374, 468512/468195
Comma list: 2200/2197, 3025/3024, 4375/4374, 468512/468195


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== Semidimfourth ==
== Semidimfourth ==
The '''semidimifourth''' temperament is featured by a semi-diminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, 235298/234375.
The '''semidimifourth''' temperament is featured by a semi-diminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, 235298/234375.
Subgroup: 2.3.5


[[Comma]]: {{monzo|7 41 -31}}
[[Comma]]: {{monzo|7 41 -31}}
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[[POTE generator]]: ~162/125 = 448.449
[[POTE generator]]: ~162/125 = 448.449


[[Vals]]: {{Val list| 8, 91, 99, 190, 289, 388, 677, 3674, 4351, 5028, 5705, 6382, 13441c, 19823bcc }}
{{Val list|legend=1| 8, 91, 99, 190, 289, 388, 677, 3674, 4351, 5028, 5705, 6382, 13441c, 19823bcc }}


[[Badness]]: 0.233376
[[Badness]]: 0.233376


=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, 235298/234375
[[Comma list]]: 4375/4374, 235298/234375


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[[POTE generator]]: ~35/27 = 448.456
[[POTE generator]]: ~35/27 = 448.456


[[Vals]]: {{Val list| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}
{{Val list|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}


[[Badness]]: 0.055249
[[Badness]]: 0.055249


=== Neusec ===
=== Neusec ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 235298/234375
Comma list: 3025/3024, 4375/4374, 235298/234375


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==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374


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== Acrokleismic ==
== Acrokleismic ==
Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, 2202927104/2197265625
[[Comma list]]: 4375/4374, 2202927104/2197265625


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[[POTE generator]]: ~6/5 = 315.557
[[POTE generator]]: ~6/5 = 315.557


[[Vals]]: {{Val list| 19, 251, 270 }}
{{Val list|legend=1| 19, 251, 270 }}


[[Badness]]: 0.056184
[[Badness]]: 0.056184


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 41503/41472, 172032/171875
Comma list: 4375/4374, 41503/41472, 172032/171875


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==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976


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=== Counteracro ===
=== Counteracro ===
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 5632/5625, 117649/117612
Comma list: 4375/4374, 5632/5625, 117649/117612


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==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374


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== Seniority ==
== Seniority ==
{{see also|Very high accuracy temperaments#Senior}}
{{see also|Very high accuracy temperaments#Senior}}
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 201768035/201326592
[[Comma list]]: 4375/4374, 201768035/201326592
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[[POTE generator]]: ~3087/2560 = 322.804
[[POTE generator]]: ~3087/2560 = 322.804


[[Vals]]: {{Val list| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}
{{Val list|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}


[[Badness]]: 0.044877
[[Badness]]: 0.044877


== Orga ==
== Orga ==
Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, 54975581388800/54936068900769
[[Comma list]]: 4375/4374, 54975581388800/54936068900769


Line 731: Line 847:
[[POTE generator]]: ~8/7 = 231.104
[[POTE generator]]: ~8/7 = 231.104


[[Vals]]: {{Val list| 26, 244, 270, 836, 1106, 1376, 2482 }}
{{Val list|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}


[[Badness]]: 0.040236
[[Badness]]: 0.040236


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 5767168/5764801
Comma list: 3025/3024, 4375/4374, 5767168/5764801


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=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360


Line 758: Line 878:


== Quatracot ==
== Quatracot ==
Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, 1483154296875/1473173782528
[[Comma list]]: 4375/4374, 1483154296875/1473173782528


Line 766: Line 888:
[[POTE generator]]: ~448/405 = 176.805
[[POTE generator]]: ~448/405 = 176.805


[[Vals]]: {{Val list| 190, 224, 414, 638, 1052c, 1690bcc }}
{{Val list|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}


[[Badness]]: 0.175982
[[Badness]]: 0.175982


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 1265625/1261568
Comma list: 3025/3024, 4375/4374, 1265625/1261568


Line 782: Line 906:


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 729/728, 1575/1573, 2200/2197
Comma list: 625/624, 729/728, 1575/1573, 2200/2197


Line 794: Line 920:
== Octoid ==
== Octoid ==
The '''octoid''' temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.
The '''octoid''' temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 16875/16807
[[Comma list]]: 4375/4374, 16875/16807
Line 810: Line 938:
* Diamond monotone and tradeoff:  [582.512, 584.359]
* Diamond monotone and tradeoff:  [582.512, 584.359]


[[Vals]]: {{Val list| 8d, 72, 152, 224 }}
{{Val list|legend=1| 8d, 72, 152, 224 }}


[[Badness]]: 0.042670
[[Badness]]: 0.042670
Line 817: Line 945:


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 4000/3993
Comma list: 540/539, 1375/1372, 4000/3993


Line 835: Line 965:


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 1375/1372, 4000/3993, 625/624
Comma list: 540/539, 1375/1372, 4000/3993, 625/624


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=== Octopus ===
=== Octopus ===
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 325/324, 364/363, 540/539
Comma list: 169/168, 325/324, 364/363, 540/539


Line 870: Line 1,004:


In the 5-limit amity is a genuine microtemperament, with 58\205 being a possible tuning. Another good choice is (64/5)<sup>1/13</sup>, which gives pure major thirds.
In the 5-limit amity is a genuine microtemperament, with 58\205 being a possible tuning. Another good choice is (64/5)<sup>1/13</sup>, which gives pure major thirds.
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 5120/5103
[[Comma list]]: 4375/4374, 5120/5103
Line 884: Line 1,020:


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4375/4374, 5120/5103
Comma list: 540/539, 4375/4374, 5120/5103


Line 895: Line 1,033:


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 540/539, 625/624, 847/845
Comma list: 352/351, 540/539, 625/624, 847/845


Line 907: Line 1,047:
=== Hitchcock ===
=== Hitchcock ===
{{see also|Amity family #Hitchcock}}
{{see also|Amity family #Hitchcock}}
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 2200/2187
Comma list: 121/120, 176/175, 2200/2187
Line 919: Line 1,061:


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 169/168, 176/175, 325/324
Comma list: 121/120, 169/168, 176/175, 325/324


Line 930: Line 1,074:


=== Hemiamity ===
=== Hemiamity ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 5120/5103
Comma list: 3025/3024, 4375/4374, 5120/5103


Line 944: Line 1,090:


In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo|8 14 -13}}, with the [[118edo|118EDO]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being {{multival|13 14 35 -8 19 42}} and adding 3136/3125 and 4375/4374, and the 11-limit wedgie {{multival|13 14 35 -36 -8 19 -102 42 -132 -222}} adding 385/384. For the 7-limit [[99edo|99EDO]] may be preferred, but in the 11-limit it is best to stick with 118.
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo|8 14 -13}}, with the [[118edo|118EDO]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7 or 11 limits, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being {{multival|13 14 35 -8 19 42}} and adding 3136/3125 and 4375/4374, and the 11-limit wedgie {{multival|13 14 35 -36 -8 19 -102 42 -132 -222}} adding 385/384. For the 7-limit [[99edo|99EDO]] may be preferred, but in the 11-limit it is best to stick with 118.
Subgroup: 2.3.5


[[Comma list]]: 1224440064/1220703125
[[Comma list]]: 1224440064/1220703125
Line 951: Line 1,099:
[[POTE generator]]: ~6/5 = 315.240
[[POTE generator]]: ~6/5 = 315.240


[[Vals]]: {{Val list| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}
{{Val list|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}


[[Badness]]: 0.043279
[[Badness]]: 0.043279


=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Comma list]]: 3136/3125, 4375/4374
[[Comma list]]: 3136/3125, 4375/4374


Line 964: Line 1,114:
[[POTE generator]]: ~6/5 = 315.181
[[POTE generator]]: ~6/5 = 315.181


[[Vals]]: {{Val list| 19, 80, 99, 217, 316, 415 }}
{{Val list|legend=1| 19, 80, 99, 217, 316, 415 }}


[[Badness]]: 0.027431
[[Badness]]: 0.027431


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 3136/3125, 4375/4374
Comma list: 385/384, 3136/3125, 4375/4374


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=== Paralytic ===
=== Paralytic ===
The ''paralytic'' temperament (118&amp;217, named by [[User:FloraC|Flora Canou]]) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118&amp;217 tempers out 1001/1000, 1575/1573, and 3584/3575.
The ''paralytic'' temperament (118&amp;217, named by [[User:FloraC|Flora Canou]]) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118&amp;217 tempers out 1001/1000, 1575/1573, and 3584/3575.
Subgroup: 2.3.5.7.11


Comma list: 441/440, 3136/3125, 4375/4374
Comma list: 441/440, 3136/3125, 4375/4374
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==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374


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==== Paraklein ====
==== Paraklein ====
The ''paraklein'' temperament (19e&amp;118, named by [[User:Xenllium|Xenllium]]) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].
The ''paraklein'' temperament (19e&amp;118, named by [[User:Xenllium|Xenllium]]) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].
Subgroup: 2.3.5.7.11.13


Comma list: 196/195, 352/351, 625/624, 729/728
Comma list: 196/195, 352/351, 625/624, 729/728
Line 1,017: Line 1,175:


=== Parkleismic ===
=== Parkleismic ===
Subgroup: 2.3.5.7.11
Comma list: 176/175, 1375/1372, 2200/2187
Comma list: 176/175, 1375/1372, 2200/2187


Line 1,028: Line 1,188:


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 176/175, 325/324, 1375/1372
Comma list: 169/168, 176/175, 325/324, 1375/1372


Line 1,039: Line 1,201:


=== Paradigmic ===
=== Paradigmic ===
Subgroup: 2.3.5.7.11
Comma list: 540/539, 896/891, 3136/3125
Comma list: 540/539, 896/891, 3136/3125


Line 1,050: Line 1,214:


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 325/324, 540/539, 832/825
Comma list: 169/168, 325/324, 540/539, 832/825


Line 1,061: Line 1,227:


=== Semiparakleismic ===
=== Semiparakleismic ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 3136/3125, 4375/4374
Comma list: 3025/3024, 3136/3125, 4375/4374


Line 1,072: Line 1,240:


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374


Line 1,083: Line 1,253:


==== Gentsemiparakleismic ====
==== Gentsemiparakleismic ====
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 325/324, 364/363, 3136/3125
Comma list: 169/168, 325/324, 364/363, 3136/3125


Line 1,097: Line 1,269:


In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo|-20 -24 25}}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as 19&amp;224 temperament (''counterkleismic'', named by [[User:Xenllium|Xenllium]]), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).
In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo|-20 -24 25}}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as 19&amp;224 temperament (''counterkleismic'', named by [[User:Xenllium|Xenllium]]), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 158203125/157351936
[[Comma list]]: 4375/4374, 158203125/157351936
Line 1,106: Line 1,280:
[[POTE generator]]: ~6/5 = 316.060
[[POTE generator]]: ~6/5 = 316.060


[[Vals]]: {{Val list| 19, 205, 224, 243, 467 }}
{{Val list|legend=1| 19, 205, 224, 243, 467 }}


[[Badness]]: 0.090553
[[Badness]]: 0.090553


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4375/4374, 2097152/2096325
Comma list: 540/539, 4375/4374, 2097152/2096325


Line 1,122: Line 1,298:


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 625/624, 729/728, 10985/10976
Comma list: 540/539, 625/624, 729/728, 10985/10976


Line 1,133: Line 1,311:


=== Counterlytic ===
=== Counterlytic ===
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 4375/4374, 496125/495616
Comma list: 1375/1372, 4375/4374, 496125/495616


Line 1,144: Line 1,324:


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 729/728, 1375/1372, 10985/10976
Comma list: 625/624, 729/728, 1375/1372, 10985/10976


Line 1,155: Line 1,337:


== Quincy ==
== Quincy ==
Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, 823543/819200
[[Comma list]]: 4375/4374, 823543/819200


Line 1,163: Line 1,347:
[[POTE generator]]: ~1728/1715 = 16.613
[[POTE generator]]: ~1728/1715 = 16.613


[[Vals]]: {{Val list| 72, 217, 289 }}
{{Val list|legend=1| 72, 217, 289 }}


[[Badness]]: 0.079657
[[Badness]]: 0.079657


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4000/3993, 4375/4374
Comma list: 441/440, 4000/3993, 4375/4374


Line 1,179: Line 1,365:


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 676/675, 4375/4374
Comma list: 364/363, 441/440, 676/675, 4375/4374


Line 1,190: Line 1,378:


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155


Line 1,201: Line 1,391:


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675


Line 1,215: Line 1,407:


Chlorine temperament has a period of 1/17 octave. It tempers out the septendecima, {{monzo|-52 -17 34}}, by which 17 chromatic semitones (25/24) exceed an octave. This temperament can be described as 289&amp;323 temperament, which tempers out {{monzo|-49 4 22 -3}} as well as the ragisma.
Chlorine temperament has a period of 1/17 octave. It tempers out the septendecima, {{monzo|-52 -17 34}}, by which 17 chromatic semitones (25/24) exceed an octave. This temperament can be described as 289&amp;323 temperament, which tempers out {{monzo|-49 4 22 -3}} as well as the ragisma.
Subgroup: 2.3.5


[[Comma]]: {{monzo|-52 -17 34}}
[[Comma]]: {{monzo|-52 -17 34}}
Line 1,222: Line 1,416:
[[POTE tuning|POTE generators]]: ~25/24 = 70.5882, ~5/4 = 386.2687
[[POTE tuning|POTE generators]]: ~25/24 = 70.5882, ~5/4 = 386.2687


[[Vals]]: {{Val list| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}
{{Val list|legend=1| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}


[[Badness]]: 0.077072
[[Badness]]: 0.077072


=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Comma list]]: 4375/4374, 193119049072265625/193091834023510016
[[Comma list]]: 4375/4374, 193119049072265625/193091834023510016


Line 1,235: Line 1,431:
[[POTE tuning|POTE generators]]: ~25/24 = 70.5882, ~5/4 = 386.2936
[[POTE tuning|POTE generators]]: ~25/24 = 70.5882, ~5/4 = 386.2936


[[Vals]]: {{Val list| 289, 323, 612, 935, 1547 }}
{{Val list|legend=1| 289, 323, 612, 935, 1547 }}


[[Badness]]: 0.041658
[[Badness]]: 0.041658


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 41503/41472, 1879453125/1879048192
Comma list: 4375/4374, 41503/41472, 1879453125/1879048192


Line 1,254: Line 1,452:


Palladium temperament has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo|-39 92 -46}}, by which 46 minortones (10/9) fall short of seven octaves. This temperament can be described as 46&amp;414 temperament, which tempers out {{monzo|-51 8 2 12}} as well as the ragisma.
Palladium temperament has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo|-39 92 -46}}, by which 46 minortones (10/9) fall short of seven octaves. This temperament can be described as 46&amp;414 temperament, which tempers out {{monzo|-51 8 2 12}} as well as the ragisma.
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 2270317133144025/2251799813685248
[[Comma list]]: 4375/4374, 2270317133144025/2251799813685248
Line 1,263: Line 1,463:
[[POTE generator]]: ~3/2 = 701.6074
[[POTE generator]]: ~3/2 = 701.6074


[[Vals]]: {{Val list| 46, 368, 414, 460, 874d }}
{{Val list|legend=1| 46, 368, 414, 460, 874d }}


[[Badness]]: 0.308505
[[Badness]]: 0.308505


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 9801/9800, 134775333/134217728
Comma list: 3025/3024, 9801/9800, 134775333/134217728


Line 1,279: Line 1,481:


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364


Line 1,290: Line 1,494:


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224


Line 1,302: Line 1,508:
== Monzism ==
== Monzism ==
The ''monzism'' temperament (53&amp;612, named by [[User:Xenllium|Xenllium]]) is a rank-two temperament which tempers out the [[monzisma]], {{monzo|54 -37 2}} and the [[nanisma]], {{monzo|109 -67 0 -1}}, as well as the ragisma, [[4375/4374]].  
The ''monzism'' temperament (53&amp;612, named by [[User:Xenllium|Xenllium]]) is a rank-two temperament which tempers out the [[monzisma]], {{monzo|54 -37 2}} and the [[nanisma]], {{monzo|109 -67 0 -1}}, as well as the ragisma, [[4375/4374]].  
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 36030948116563575/36028797018963968
[[Comma list]]: 4375/4374, 36030948116563575/36028797018963968
Line 1,311: Line 1,519:
[[POTE generator]]: ~310078125/268435456 = 249.0207
[[POTE generator]]: ~310078125/268435456 = 249.0207


[[Vals]]: {{Val list| 53, 559, 612, 1277, 1889 }}
{{Val list|legend=1| 53, 559, 612, 1277, 1889 }}


[[Badness]]: 0.046569
[[Badness]]: 0.046569


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 41503/41472, 184549376/184528125
Comma list: 4375/4374, 41503/41472, 184549376/184528125


Line 1,327: Line 1,537:


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625


Retrieved from "https://en.xen.wiki/w/Ragismic_microtemperaments"