Ragismic microtemperaments: Difference between revisions
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Since {{nowrap|(10/9)<sup>4</sup> {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)<sup>2</sup> }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal. | Since {{nowrap|(10/9)<sup>4</sup> {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)<sup>2</sup> }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal. | ||
Temperaments discussed elsewhere are: | |||
* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]] | |||
* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]] | |||
* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]] | |||
* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]] | * ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]] | ||
* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]] | |||
* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]] | |||
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]] | |||
* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]] | |||
* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]] | |||
* [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]] | |||
* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]] | |||
* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]] | |||
* ''[[Zarvo]]'' (+33075/32768) → [[Gravity family #Zarvo|Gravity family]] | |||
* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]] | |||
* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]] | * ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]] | ||
* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]] | |||
* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]] | * ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]] | ||
* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]] | * ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]] | ||
* | * [[Parakleismic]] (+3136/3125) → [[Parakleismic family #Septimal parakleismic|Parakleismic family]] | ||
* [[ | * ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnu family #Septimal vishnu|Vishnu family]] | ||
* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]] | * ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]] | ||
* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortone family #Mitonic|Minortone family]] | |||
* ''[[Mitonic]]'' (+2100875/2097152) → [[ | |||
* ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]] | * ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]] | ||
* | * [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]] | ||
* ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments #Dzelic|37th-octave temperaments]] | * ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments #Dzelic|37th-octave temperaments]] | ||
Considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, chlorine, octoid, seniority, monzismic, semidimfourth, acrokleismic, quasithird, quincy, deca, keenanose, counterkleismic, sfourth, aluminium, ragitritonic, quatracot, trideci, moulin, and palladium. | |||
== Supermajor == | == Supermajor == | ||
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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 }}). | The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}), and may be described as the {{nowrap| 217 & 224 }} temperament. | ||
Early in the design of the [[Sagittal]] notation system, [[George Secor|Secor]] and [[Dave Keenan|Keenan]] found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4{{c}} many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of brahmagupta that has pure octaves and pure fifths, which can also be described as a 17-limit extension having a 1/7-octave period (171.4286{{c}}) and 1/21-apotome generator (5.4136{{c}}). | Early in the design of the [[Sagittal]] notation system, [[George Secor|Secor]] and [[Dave Keenan|Keenan]] found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4{{c}} many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of brahmagupta that has pure octaves and pure fifths, which can also be described as a 17-limit extension having a 1/7-octave period (171.4286{{c}}) and 1/21-apotome generator (5.4136{{c}}). | ||
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: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Abigail]].'' | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Abigail]].'' | ||
Abigail tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by [[Gene Ward Smith]] in 2010 after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930 Yahoo! Tuning Group | ''11-limit rank 2 using only wedgies''] "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things." —Gene Ward Smith</ref> | Abigail tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit, and may be described as the {{nowrap| 46 & 224 }} temperament, with a [[ploidacot]] signature of diploid wau-hendecacot. It extends into a very strong 11- and 13-limit temperament. [[494edo]], [[764edo]] and [[1258edo]] are among the possible tunings. | ||
Abigail was named by [[Gene Ward Smith]] in 2010 after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930 Yahoo! Tuning Group | ''11-limit rank 2 using only wedgies''] "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things." —Gene Ward Smith</ref> | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].'' | : ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].'' | ||
Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament. [[1106edo]] | Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament, with a [[ploidacot]] of diploid alpha-octacot. [[1106edo]] gives a strong tuning. | ||
Crazy was named by [[Flora Canou]] in 2025 by removing the mutation from ''kwazy'', the name for the 5-limit microtemperament. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }} | {{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }} | ||
: | : mapping generators: ~332150625/234881024, ~1125/1024 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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== Orga == | == Orga == | ||
Orga may be described as the {{nowrap| 26 & 270 }} temperament, and [[1106edo]] gives a strong tuning. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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Badness (Sintel): 0.899 | Badness (Sintel): 0.899 | ||
== | == Chlorine == | ||
: ''For the 5-limit version, see [[ | : ''For the 5-limit version, see [[17th-octave temperaments #Chlorine]].'' | ||
Chlorine (named after the 17th element) tempers out the [[septendecima]] in the 5-limit, and {{monzo| -49 4 22 -3 }} as well as the ragisma in the 7-limit. It has a 1/17-octave period, and can be described as {{nowrap| 289 & 323 }} temperament. Not only the semitwelfth, but also the ~5/4 can be used as a generator. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, | [[Comma list]]: 4375/4374, {{monzo| -49 4 22 -3 }} | ||
{{Mapping|legend=1| | {{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~ | * [[WE]]: ~25/24 = 70.5880{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9962{{c}} | ||
: [[error map]]: {{val| | : [[error map]]: {{val| -0.004 +0.037 -0.030 -0.019 }} | ||
* [[CWE]]: ~ | * [[CWE]]: ~25/24 = 70.5882{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9990{{c}} | ||
: error map: {{val| 0.000 | : error map: {{val| 0.000 +0.043 -0.021 -0.013 }} | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547, 3706, 5253 }} | ||
[[Badness]] (Sintel): 1. | [[Badness]] (Sintel): 1.05 | ||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 4375/4374, 41503/41472, 1879453125/1879048192 | ||
Mapping: {{mapping| | Mapping: {{mapping| 17 0 26 -87 207 | 0 2 1 10 -11 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~25/24 = 70.5905{{c}}, ~693/400 = 951.0054{{c}} | ||
* CWE: ~ | * CWE: ~25/24 = 70.5882{{c}}, ~693/400 = 950.9754{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 289, 323, 612, 3349de, 3961de, …, 5797ddee }} | ||
Badness (Sintel): | Badness (Sintel): 2.11 | ||
==== | == Octoid == | ||
: {{Main| Octoid }} | |||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #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 comma]]). In the 11-limit, it tempers out [[540/539]], [[1375/1372]], and [[6250/6237]]. In this temperament, one period gives ~[[12/11]], two give ~[[25/21]], three give ~[[35/27]], and four give [[99/70]]~[[140/99]]. | |||
The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimizing the average damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out [[324/323]], [[375/374]], [[495/494]], [[625/624]], [[715/714]] or [[729/728]]. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 4375/4374, 16875/16807 | |||
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }} | |||
: mapping generators: ~49/45, ~7/5 | |||
== | [[Optimal tuning]]s: | ||
* [[WE]]: ~49/45 = 150.0003{{c}}, ~7/5 = 583.9416{{c}} | |||
: [[error map]]: {{val| +0.002 -0.130 -0.547 +0.883 }} | |||
* [[CWE]]: ~49/45 = 150.0000{{c}}, ~7/5 = 583.9411{{c}} | |||
: error map: {{val| 0.000 -0.132 -0.549 +0.880 }} | |||
[[Tuning ranges]]: | |||
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8) | |||
* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88) | |||
* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359] | |||
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084] | |||
{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }} | |||
[[Badness]] (Sintel): 1.08 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 540/539, 1375/1372, 4000/3993 | |||
Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }} | |||
: | |||
Optimal tunings: | |||
* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}} | |||
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9477{{c}} | |||
[ | Tuning ranges: | ||
* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88) | |||
* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084] | |||
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224, 824d }} | |||
Badness (Sintel): 0.466 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 540/539, 625/624, 729/728, 1375/1372 | |||
Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }} | |||
Mapping: {{mapping| 1 | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}} | ||
* CWE: ~ | * CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 72, 152f, 224 }} | ||
Badness (Sintel): | Badness (Sintel): 0.631 | ||
==== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 375/374, 540/539, 625/624, 715/714, 729/728 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}} | ||
* CWE: ~ | * CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }} | ||
Badness (Sintel): | Badness (Sintel): 0.729 | ||
== | ===== 19-limit ===== | ||
: | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714 | |||
Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }} | |||
Optimal tunings: | |||
* WE: ~12/11 = 149.9785{{c}}, ~7/5 = 583.8482{{c}} | |||
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 72, 152fg, 224 }} | ||
Badness (Sintel): 0.975 | |||
{{ | ==== Octopus ==== | ||
A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4 = 300{{c}}. | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 169/168, 325/324, 364/363, 540/539 | |||
Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }} | |||
Mapping: {{mapping| | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~12/11 = 150.0313{{c}}, ~7/5 = 584.0134{{c}} | ||
* CWE: ~ | * CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}} | ||
{{Optimal ET sequence|legend=0| 8d, …, | {{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }} | ||
Badness (Sintel): | Badness (Sintel): 0.896 | ||
==== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 169/168, 221/220, 289/288, 325/324, 540/539 | ||
Mapping: {{mapping| | Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}} | ||
* CWE: ~ | * CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}} | ||
{{Optimal ET sequence|legend=0| 8d, …, | {{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }} | ||
Badness (Sintel): | Badness (Sintel): 0.795 | ||
== | ===== 19-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399 | |||
{{ | Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }} | ||
Optimal tunings: | |||
* | * WE: ~12/11 = 150.0049{{c}}, ~7/5 = 584.0833{{c}} | ||
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}} | |||
* | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 8d, 72, 152 }} | ||
Badness (Sintel): 0.993 | |||
Scales: [[Octoid72]], [[Octoid80]] | |||
==== Hexadecoid ==== | |||
{{See also| 16th-octave temperaments }} | |||
Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224. | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224 | |||
Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }} | |||
: mapping generators: ~448/429, ~7/5 | |||
== | Optimal tunings: | ||
* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}} | |||
* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}} | |||
{{Optimal ET sequence|legend=0| 80, 144, 224 }} | |||
Badness (Sintel): 1.27 | |||
===== 17-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224 | |||
Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }} | |||
== | Optimal tunings: | ||
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9626{{c}} | |||
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0463{{c}} | |||
{{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }} | |||
Badness (Sintel): 1.46 | |||
===== 19-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444 | |||
Comma list: | |||
Mapping: {{mapping| 1 - | Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}} | ||
* CWE: ~ | * CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.44 | ||
== | == Seniority == | ||
: ''For the 5-limit version, see [[ | : ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]]. | ||
Aside from the ragisma, the seniority temperament tempers out the [[wadisma]], 201768035/201326592, and may be described as {{nowrap| 26 & 145 }}. It is so named because the [[senior comma]] ({{monzo| -17 62 -35 }}) is tempered out. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, | [[Comma list]]: 4375/4374, 201768035/201326592 | ||
{{Mapping|legend=1| | {{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }} | ||
: mapping generators: ~2, ~5120/3087 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.0745{{c}}, ~5120/3087 = 877.2500{{c}} | |||
: [[error map]]: {{val| +0.075 +0.008 -0.016 -0.203 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5120/3087 = 877.1965{{c}} | |||
: error map: {{val| 0.000 -0.077 -0.130 -0.415 }} | |||
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d }} | |||
[[Badness]] (Sintel): 1.14 | |||
[[ | === Senator === | ||
Senator (26 & 145) extends seniority by tempering out [[441/440]] and [[65536/65219]], and can be extended to the 13- and 17-limit immediately by adding [[364/363]] and [[595/594]] to the comma list in this order. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 441/440, 4375/4374, 65536/65219 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 -24 -43 5 2 | 0 35 62 -3 2 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e }} | ||
Badness (Sintel): | Badness (Sintel): 3.05 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 364/363, 441/440, 2200/2197, 4375/4374 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.7136{{c}}, ~108/65 = 877.9974{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2038{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.85 | ||
== | ==== 17-limit ==== | ||
: | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197 | |||
Mapping: {{mapping| 1 -24 -43 5 2 -27 -31 | 0 35 62 -3 2 42 48 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.7195{{c}}, ~108/65 = 877.0018{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }} | ||
Badness (Sintel): 1.35 | |||
== Monzismic == | |||
: ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]]. | |||
[[ | Monzismic tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]]. It may be described as the {{nowrap| 53 & 612 }} temperament, with a [[ploidacot]] signature of alpha-dicot. A notable tuning not appearing on the optimal ET sequence is [[665edo]], which is nearly equivalent to the pure-3's tuning. | ||
[[Subgroup]]: 2.3.5.7 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }} | ||
Mapping: {{ | {{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }} | ||
: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }} | |||
Optimal tuning | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.0128{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9895{{c}} | |||
: [[error map]]: {{val| +0.013 +0.024 -0.049 -0.019 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9793{{c}} | |||
: error map: {{val| 0.000 +0.004 -0.080 -0.050 }} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd }} | ||
Badness (Sintel): | [[Badness]] (Sintel): 1.18 | ||
=== | === Monzism === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 4375/4374, 41503/41472, 184549376/184528125 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 0 -27 109 -159 | 0 2 37 -134 205 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}} | |||
{{Optimal ET sequence|legend=0| 53, 559, 612, 3619de, 4231de, …, 6067ddee }} | |||
Badness (Sintel): 1.89 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625 | |||
Mapping: {{mapping| 1 0 -27 109 -159 -70 | 0 2 37 -134 205 93 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 53, 559, 612 }} | ||
Badness (Sintel): 2.22 | |||
== Semidimfourth == | |||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimfourth]].'' | |||
The semidimfourth temperament is featured by a semidiminished 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]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, | [[Comma list]]: 4375/4374, 235298/234375 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 -10 -13 -17 | 0 31 41 53 }} | ||
: mapping generators: ~2, ~ | : mapping generators: ~2, ~35/27 | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1199.9936{{c}}, ~35/27 = 448.4533{{c}} | |||
: [[error map]]: {{val| -0.007 +0.160 +0.353 -0.694 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 448.4555{{c}} | |||
: error map: {{val| 0.000 +0.165 +0.361 -0.685 }} | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 8d, …, 91, 99, 289, 388, 875 }} | ||
[[Badness]] (Sintel): | [[Badness]] (Sintel): 1.40 | ||
=== | === Neusec === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 4375/4374, | Comma list: 3025/3024, 4375/4374, 235298/234375 | ||
Mapping: {{mapping| | Mapping: {{mapping| 2 -20 -26 -34 -17 | 0 31 41 53 32 }} | ||
: mapping generators: ~99/70, ~35/27 | |||
Optimal | Optimal tunings: | ||
* WE: ~99/70 = 600.0381{{c}}, ~35/27 = 448.4812{{c}} | |||
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4546{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 8d, …, 190, 388 }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.95 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374 | ||
Mapping: {{mapping| | Mapping: {{mapping| 2 -20 -26 -34 -17 -21 | 0 31 41 53 32 38 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~99/70 = 600.0034{{c}}, ~35/27 = 448.4573{{c}} | |||
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4549{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 8d, …, 190, 198, 388 }} | ||
Badness (Sintel): | Badness (Sintel): 1.28 | ||
== Acrokleismic == | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, | [[Comma list]]: 4375/4374, 2202927104/2197265625 | ||
{{Mapping|legend=1| 1 -22 -22 -65 | 0 32 33 92 }} | |||
: | : mapping generators: ~2, ~5/3 | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1199.9305{{c}}, ~5/3 = 884.3923{{c}} | |||
: [[error map]]: {{val| -0.070 +0.126 +0.160 -0.221 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.4423{{c}} | |||
: error map: {{val| 0.000 +0.198 +0.282 -0.136 }} | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }} | ||
[[Badness]] (Sintel): | [[Badness]] (Sintel): 1.42 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 4375/4374, | Comma list: 4375/4374, 41503/41472, 172032/171875 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 -22 -22 -65 58 | 0 32 33 92 -74 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9698{{c}}, ~5/3 = 884.4193{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4414{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 19, 251, 270, 829, 1099, 1369, 1639 }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.22 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 -22 -22 -65 58 -56 | 0 32 33 92 -74 81 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9939{{c}}, ~5/3 = 884.4384{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4429{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 19, 251, 270 }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.11 | ||
== | === Counteracro === | ||
: | Subgroup: 2.3.5.7.11 | ||
Comma list: 4375/4374, 5632/5625, 117649/117612 | |||
Mapping: {{mapping| 1 -22 -22 -65 -141 | 0 32 33 92 196 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.8877{{c}}, ~5/3 = 884.3639{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4457{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1061e, 1331c, 1601c, 1871bc }} | ||
Badness (Sintel): 1.41 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374 | |||
Mapping: {{mapping| 1 -22 -22 -65 -141 -56 | 0 32 33 92 196 81 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.9285{{c}}, ~5/3 = 884.3937{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4458{{c}} | |||
{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1331c }} | |||
Badness (Sintel): 1.08 | |||
== Quasithird == | |||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasithird]].'' | |||
Quasithird may be described as the {{nowrap| 224 & 388 }} temperament, 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 it to temper out the ragisma and {{monzo| -60 29 0 5 }}. Its [[ploidacot]] is tetraploid delta-pentacot. | |||
{{ | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, {{monzo| - | [[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }} | ||
{{Mapping|legend=1| | {{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }} | ||
: mapping generators: ~ | : mapping generators: ~65536/55125, ~5103/4096 | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~65536/55125 = 300.0052{{c}}, ~5103/4096 = 380.3949{{c}} | |||
: [[error map]]: {{val| +0.021 +0.020 -0.052 -0.031 }} | |||
* [[CWE]]: ~65536/55125 = 300.0000{{c}}, ~5103/4096 = 380.3884{{c}} | |||
: error map: {{val| 0.000 -0.013 -0.100 -0.089 }} | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }} | ||
[[Badness]] (Sintel): | [[Badness]] (Sintel): 1.56 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 3025/3024, 4375/4374, | Comma list: 3025/3024, 4375/4374, 4296700485/4294967296 | ||
Mapping: {{mapping| | Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~65536/51125 = 300.0073{{c}}, ~5103/4096 = 380.3963{{c}} (or ~22/21 = 80.3890{{c}}) | |||
* CWE: ~65536/51125 = 300.0000{{c}}, ~5103/4096 = 380.3868{{c}} (or ~22/21 = 80.3868{{c}}) | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee }} | ||
Badness (Sintel): | Badness (Sintel): 0.698 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374 | ||
Mapping: {{mapping| | Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~65536/51125 = 299.9985{{c}}, ~81/65 = 380.3833{{c}} (or ~22/21 = 80.3848{{c}}) | |||
* CWE: ~65536/51125 = 300.0000{{c}}, ~81/65 = 380.3852{{c}} (or ~22/21 = 80.3852{{c}}) | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836 }} | ||
Badness (Sintel): | Badness (Sintel): 1.22 | ||
== Quincy == | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, | [[Comma list]]: 4375/4374, 823543/819200 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }} | ||
: | : mapping generators: ~2, ~1728/1715 | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.2169{{c}}, ~1728/1715 = 16.6160{{c}} | |||
: [[error map]]: {{val| +0.217 +0.000 +0.155 -0.799 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1728/1715 = 16.6083{{c}} | |||
: error map: {{val| 0.000 -0.205 -0.122 -1.343 }} | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 72, 217, 289, 650d, 939dd }} | ||
[[Badness]] (Sintel): | [[Badness]] (Sintel): 2.02 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 441/440, 4000/3993, 4375/4374 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.1286{{c}}, ~100/99 = 16.6147{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6101{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 72, 217, 289 }} | ||
Badness (Sintel): | Badness (Sintel): 1.02 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 364/363, 441/440, 676/675, 4375/4374 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.0554{{c}}, ~100/99 = 16.6028{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6011{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 72, 145, 217, 289 }} | ||
Badness (Sintel): | Badness (Sintel): 0.986 | ||
== | === 17-limit === | ||
: | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155 | |||
Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.0647{{c}}, ~100/99 = 16.6025{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6004{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 72, 145, 217, 289 }} | ||
Badness (Sintel): 0.751 | |||
=== 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 | |||
Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.9287{{c}}, ~100/99 = 16.5930{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.5948{{c}} | |||
{{Optimal ET sequence|legend=0| 72, 145, 217 }} | |||
Badness (Sintel): 0.924 | |||
== Deca == | |||
: ''For 5-limit version, see [[10th-octave temperaments#Neon]].'' | |||
Deca has a period of 1/10 octave and tempers out the [[neon comma]] ({{monzo| 21 60 -50 }}) in the 5-limit, the [[linus comma]] ({{monzo| 11 -10 -10 10 }}) and {{monzo| 12 -3 -14 9 }} (165288374272/164794921875) in the 7-limit. It may be described as the {{nowrap| 80 & 190 }} temperament, and has a [[ploidacot]] of decaploid wau-pentacot. | |||
[[Subgroup]]: 2.3.5.7 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[Comma list]]: 4375/4374, 165288374272/164794921875 | ||
{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }} | |||
: mapping generators: ~15/14, ~460992/390625 | |||
Optimal tuning ( | [[Optimal tuning]]s: | ||
* [[WE]]: ~15/14 = 119.9966{{c}}, ~460992/390625 = 284.4150{{c}} (5625/5488 = 44.4219{{c}}) | |||
: [[error map]]: {{val| -0.034 +0.106 +0.145 -0.268 }} | |||
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~460992/390625 = 284.4182{{c}} (5625/5488 = 44.4182{{c}}) | |||
: error map: {{val| 0.000 +0.136 +0.195 -0.226 }} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }} | ||
Badness (Sintel): | [[Badness]] (Sintel): 2.04 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 3025/3024, 4375/4374, 391314/390625 | ||
Mapping: {{mapping| | Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~15/14 = 120.0004{{c}}, ~33/28 = 284.4193{{c}} (77/75 = 44.4185{{c}}) | |||
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4189{{c}} (77/75 = 44.4189{{c}}) | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 80, 190, 270, 1000, 1270, 1540e, 1810e }} | ||
Badness (Sintel): | Badness (Sintel): 0.804 | ||
== | === 13-limit === | ||
: | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374 | |||
Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }} | |||
Optimal tunings: | |||
* WE: ~15/14 = 120.0067{{c}}, ~33/28 = 284.4139{{c}} (~40/39 = 44.4006{{c}}) | |||
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4048{{c}} (~40/39 = 44.4048{{c}}) | |||
{{ | {{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }} | ||
Badness (Sintel): 0.695 | |||
=== 2.3.5.7.11.13.19 subgroup === | |||
Subgroup: 2.3.5.7.11.13.19 | |||
Comma list: 1001/1000, 1521/1520, 3025/3024, 4225/4224, 4375/4374 | |||
Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }} | |||
: | |||
Optimal tunings: | |||
* WE: ~15/14 = 120.0045{{c}}, ~33/28 = 284.4140{{c}} (~39/38 = 44.4050{{c}}) | |||
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4075{{c}} (~39/38 = 44.4075{{c}}) | |||
{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }} | |||
Badness (Sintel): 0.556 | |||
[[ | == Keenanose == | ||
Keenanose, the {{nowrap| 270 & 1889 }} temperament, was named by [[Eliora]] in 2022 for the fact that it uses [[385/384]], the keenanisma, as the generator. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[ | [[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }} | ||
{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }} | |||
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }} | |||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.0068{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4467{{c}} | |||
: [[error map]]: {{val| +0.007 +0.031 -0.035 -0.032 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4466{{c}} | |||
: error map: {{val| 0.000 +0.025 -0.043 -0.050 }} | |||
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }} | |||
[[Badness]] (Sintel): 2.17 | |||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 4375/4374, 117649/117612, 67110351/67108864 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9970{{c}}, ~385/384 = 4.4465{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4465{{c}} | |||
{{Optimal ET sequence|legend=0| 270, 1349, 1619, 1889, 2159, 11065, 13224 }} | |||
Badness (Sintel): 1.02 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612 | |||
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.0065{{c}}, ~385/384 = 4.4467{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4467{{c}} | |||
{{Optimal ET sequence|legend=0| 270, 1079, 1349, 1619, 1889, 4048 }} | |||
Badness (Sintel): 0.879 | |||
== Counterkleismic == | |||
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].'' | |||
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)<sup>6</sup>) fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament, tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). It was named by analogy to [[catakleismic]] and [[parakleismic]]). | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 4375/4374, 158203125/157351936 | |||
= | {{Mapping|legend=1| 1 -5 -4 -18 | 0 25 24 79 }} | ||
: mapping generators: ~2, ~6/5 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.1778{{c}}, ~6/5 = 316.1065{{c}} | |||
: [[error map]]: {{val| +0.178 -0.181 -0.469 +0.388 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.0631{{c}} | |||
: error map: {{val| 0.000 -0.377 -0.799 +0.161 }} | |||
{{Optimal ET sequence|legend=1| 19, …, 205, 224, 243, 467 }} | |||
[[Badness]] (Sintel): 2.29 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 540/539, 4375/4374, 2097152/2096325 | |||
Mapping: {{mapping| 1 -5 -4 -18 19 | 0 25 24 79 -59 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.9944{{c}}, ~6/5 = 316.0690{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0705{{c}} | |||
{{Optimal ET sequence|legend=0| 19, 205, 224 }} | |||
Badness (Sintel): 2.35 | |||
Badness (Sintel): | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 540/539, 625/624, 729/728, 10985/10976 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 -5 -4 -18 19 -15 | 0 25 24 79 -59 71 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9827{{c}}, ~6/5 = 316.0650{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0695{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 19, 205, 224 }} | ||
Badness (Sintel): | Badness (Sintel): 1.40 | ||
=== | === Counterlytic === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 1375/1372, 4375/4374, 496125/495616 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 -5 -4 -18 -40 | 0 25 24 79 165 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.1247{{c}}, ~6/5 = 316.0976{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0660{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=1| 19e, 205e, 224, 467e, 691, 915c }} | ||
Badness (Sintel): | Badness (Sintel): 2.16 | ||
==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 625/624, 729/728, 1375/1372, 10985/10976 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 -5 -4 -18 -40 -15 | 0 25 24 79 165 71 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.0987{{c}}, ~6/5 = 316.0908{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0658{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 19e, 205e, 224, 467e, 691, 915c }} | ||
Badness (Sintel): | Badness (Sintel): 1.23 | ||
== Sfourth == | |||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sfourth]].'' | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 4375/4374, 64827/64000 | |||
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }} | |||
: mapping generators: ~2, ~49/48 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.8332{{c}}, ~49/48 = 26.3053{{c}} | |||
: [[error map]]: {{val| +0.833 -0.090 +0.721 -3.074 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 26.2590{{c}} | |||
: error map: {{val| 0.000 -0.876 -0.343 -5.157 }} | |||
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }} | |||
[[Badness]] (Sintel): 3.12 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 121/120, 441/440, 4375/4374 | |||
Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1201.1486{{c}}, ~49/48 = 26.3112{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2461{{c}} | |||
{{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }} | |||
Badness (Sintel): 1.78 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 121/120, 169/168, 325/324, 441/440 | |||
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1201.4956{{c}}, ~49/48 = 26.3423{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2614{{c}} | |||
{{Optimal ET sequence|legend=0| 45ef, 46, 91ef, 137def, 228ddeeefff }} | |||
Badness (Sintel): 1.37 | |||
=== Sfour === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 385/384, 2401/2376, 4375/4374 | |||
Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }} | |||
{{ | |||
Optimal tunings: | |||
* WE: ~2 = 1200.4402{{c}}, ~49/48 = 26.2557{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2403{{c}} | |||
{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }} | |||
Badness (Sintel): 2.53 | |||
==== 13-limit ==== | |||
: | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 196/195, 364/363, 385/384, 4375/4374 | |||
{{ | Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1200.3796{{c}}, ~49/48 = 26.2473{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2372{{c}} | |||
= | {{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }} | ||
Badness (Sintel): 2.14 | |||
== Aluminium == | |||
: ''For the 5-limit version, see [[13th-octave temperaments #Aluminium]].'' | |||
Aluminium tempers out {{monzo| 92 -39 -13 }} in the 5-limit and sets [[135/128]] to 1/13 of an [[octave]]. It was named by [[Eliora]] in 2023 after the 13th element. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }} | |||
[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }} | |||
: Mapping generators: ~135/128, ~3 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~135/128 = 92.3072{{c}}, ~3/2 = 701.9995{{c}} | |||
: [[error map]]: {{val| -0.006 +0.038 -0.030 -0.013 }} | |||
* [[CWE]]: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0030{{c}} | |||
: error map: {{val| 0.000 +0.048 -0.015 +0.001 }} | |||
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }} | |||
[[Badness]] (Sintel): 3.20 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 4375/4374, 234375/234256, 2097152/2096325 | |||
Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }} | |||
== | Optimal tunings: | ||
* WE: ~135/128 = 92.3062{{c}}, ~3/2 = 701.9946{{c}} | |||
* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0056{{c}} | |||
{{Optimal ET sequence|legend=0| 494, 1053, 1547, 3588e, 5135e }} | |||
Badness (Sintel): 1.39 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078 | |||
Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }} | |||
Optimal tunings: | |||
* WE: ~135/128 = 92.3055{{c}}, ~3/2 = 701.9928{{c}} | |||
* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0098{{c}} | |||
{{Optimal ET sequence|legend=0| 494, 1547, 2041, 4576def }} | |||
Badness (Sintel): 1.18 | |||
== Ragitritonic == | |||
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].'' | |||
Ragitritonic may be described as the {{nowrap| 53 & 369 }} temperament, splitting the [[24/1|24th harmonic]] into nine tritone generators; its [[ploidacot]] is thus delta-enneacot. [[422edo]] makes for a strong tuning. | |||
Ragitritonic was named by [[Flora Canou]] in 2026 as a contraction of ''ragismic'' and ''tritonic''. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 4375/4374, 68719476736/68356598625 | |||
{{Mapping|legend=1| 1 -3 -15 40 | 0 9 34 -73 }} | |||
: mapping generators: ~2, ~65536/45927 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.8189{{c}}, ~65536/45927 = 611.2850{{c}} | |||
: [[error map]]: {{val| -0.181 +0.153 +0.094 +0.123 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~65536/45927 = 611.3775{{c}} | |||
: error map: {{val| 0.000 +0.443 +0.522 +0.615 }} | |||
Optimal | {{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }} | ||
[[Badness]] (Sintel): 3.37 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 4375/4374, 5632/5625, 2621440/2614689 | |||
Mapping: {{mapping| 1 -3 -15 40 -75 | 0 9 34 -73 154 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.8147{{c}}, ~768/539 = 611.2822{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~768/539 = 611.3762{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 53, 316e, 369, 422, 791e, 1213cde }} | ||
Badness (Sintel): 2.34 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625 | |||
Mapping: {{mapping| 1 -3 -15 40 -75 -34 | 0 9 34 -73 154 74 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.7916{{c}}, ~91/64 = 611.2698{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~91/64 = 611.3754{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }} | ||
Badness (Sintel): 1.51 | |||
== Quatracot == | |||
{{See also| Stratosphere }} | |||
[[Subgroup]]: 2.3.5.7 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }} | ||
{{Mapping|legend=1| 2 -6 -1 -36 | 0 13 8 59 }} | |||
: mapping generators: ~2278125/1605632, ~7168/5625 | |||
Optimal tuning | [[Optimal tuning]]s: | ||
* [[WE]]: ~2278125/1605632 = 600.0888{{c}}, ~7168/5625 = 423.2574{{c}} | |||
: [[error map]]: {{val| +0.178 -0.141 -0.343 +0.165 }} | |||
* [[CWE]]: ~2278125/1605632 = 600.0000{{c}}, ~7168/5625 = 423.1986{{c}} | |||
: error map: {{val| 0.000 -0.374 -0.725 -0.111 }} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=1| 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc }} | ||
Badness (Sintel): | [[Badness]] (Sintel): 4.45 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 3025/3024, | Comma list: 3025/3024, 4375/4374, 1265625/1261568 | ||
Mapping: {{mapping| 2 | Mapping: {{mapping| 2 -6 -1 -36 -22 | 0 13 8 59 41 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~99/70 = 600.0847{{c}}, ~225/176 = 423.2536{{c}} | |||
* CWE: ~99/70 = 600.0000{{c}}, ~225/176 = 423.1977{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638, 1052c }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.36 | ||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 625/624, 729/728, 1575/1573, 2200/2197 | ||
Mapping: {{mapping| 2 | Mapping: {{mapping| 2 -6 -1 -36 -22 -6 | 0 13 8 59 41 19 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~99/70 = 600.0571{{c}}, ~143/112 = 423.2366{{c}} | |||
* CWE: ~99/70 = 600.0000{{c}}, ~143/112 = 423.1987{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638 }} | ||
Badness (Sintel): | Badness (Sintel): 0.936 | ||
== | == Trideci == | ||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tridecatonic]].'' | |||
The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic]] temperament, but with the ragisma (4375/4374) rather than the octagar comma (4000/3969) tempered out. The name ''trideci'' comes from ''tridecim'' (Latin for "thirteen"). | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 4375/4374, 83349/81920 | |||
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }} | |||
: mapping generators: ~256/245, ~3 | |||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~256/245 = 92.4141{{c}}, ~3/2 = 699.9466{{c}} | |||
: [[error map]]: {{val| +1.383 -0.626 -0.210 -2.554 }} | |||
* [[CWE]]: ~256/245 = 92.3077{{c}}, ~3/2 = 699.4521{{c}} | |||
: error map: {{val| 0.000 -2.503 -2.794 -6.740 }} | |||
{{Optimal ET sequence|legend=1| 26, 65, 91 }} | |||
[[Badness]] (Sintel): 4.67 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 245/242, 385/384, 4375/4374 | |||
Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }} | |||
{{ | Optimal tunings: | ||
* WE: ~22/21 = 92.3729{{c}}, ~3/2 = 700.1118{{c}} | |||
* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.7703{{c}} | |||
{{Optimal ET sequence|legend=0| 26, 65, 91 }} | |||
Badness (Sintel): 2.80 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 169/168, 245/242, 325/324, 385/384 | |||
Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }} | |||
Optimal tunings: | |||
* WE: ~22/21 = 92.4003{{c}}, ~3/2 = 699.9983{{c}} | |||
* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.4772{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 26, 65f, 91f }} | ||
Badness (Sintel): 2.16 | |||
== Moulin == | |||
Moulin can be described as the {{nowrap| 494 & 1619 }} temperament. It has a generator of ~[[22/13]], and it was named by [[Eliora]] in 2022 after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". However, the functional generator is ~[[13/11]], and 73 of them octave reduced reach the [[3/2|perfect fifth]]. Since [[11/8]] is within 23 generators, the 25-tone generator chain (4L 21s) of this temperament contains the 8:11:13 triad. | |||
[[Subgroup]]: 2.3.5.7 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }} | ||
{{Mapping|legend=1| 1 -16 -9 -75 | 0 73 47 323 }} | |||
: mapping generators: ~2, ~3796875/3211264 | |||
Optimal tuning | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.0272{{c}}, ~3796875/3211264 = 289.0675{{c}} | |||
: [[error map]]: {{val| +0.027 +0.007 -0.084 +0.013 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3796875/3211264 = 289.0675{{c}} | |||
: error map: {{val| 0.000 -0.029 -0.142 -0.029 }} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=1| 494, 1125, 1619, 8589cc, 10208cc }} | ||
Badness (Sintel): | [[Badness]] (Sintel): 5.93 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 4375/4374, 759375/758912, 100663296/100656875 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -16 -9 -75 9 | 0 73 47 323 -23 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.0043{{c}}, ~605/512 = 289.0687{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~605/512 = 289.0677{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }} | ||
Badness (Sintel): 2. | Badness (Sintel): 2.24 | ||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078 | ||
Mapping: {{mapping| 1 -16 -9 -75 9 9 | 0 73 47 323 -23 -22 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.0043{{c}}, ~13/11 = 289.0687{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 289.0677{{c}} | |||
{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }} | |||
Badness (Sintel): 1.12 | |||
== Palladium == | |||
: ''For the 5-limit version, see [[46th-octave temperaments #Palladium]]''. | |||
The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minor whole tones (10/9) fall short of seven octaves. This temperament can be described as {{nowrap| 46 & 414 }} temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 4375/4374, | [[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }} | ||
{{Mapping|legend=1| 1 2 | {{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }} | ||
: mapping generators: ~83349/81920, ~3 | |||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~83349/81920 = 26.0910{{c}}, ~3/2 = 701.7155{{c}} | |||
: [[error map]]: {{val| +0.185 -0.055 -0.061 +0.349 }} | |||
* [[CWE]]: ~83349/81920 = 26.0870{{c}}, ~3/2 = 701.6491{{c}} | |||
: error map: {{val| 0.000 -0.306 -0.407 -0.910 }} | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 46, …, 368, 414, 460, 874d }} | ||
[[Badness]] (Sintel): | [[Badness]] (Sintel): 7.81 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 3025/3024, 4375/4374, 134775333/134217728 | ||
Mapping: {{mapping| 1 2 | Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~8192/8085 = 26.0912{{c}}, ~3/2 = 701.7082{{c}} | |||
* CWE: ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.6173{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 46, …, 368, 414, 460, 874de }} | ||
Badness (Sintel): | Badness (Sintel): 2.44 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364 | ||
Mapping: {{mapping| 1 2 | Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7411{{c}} | |||
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6465{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334dde }} | ||
Badness (Sintel): | Badness (Sintel): 1.68 | ||
=== 17-limit === | === 17-limit === | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224 | ||
Mapping: {{mapping| 1 2 | Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7399{{c}} | |||
{{ | * CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6464{{c}} | ||
{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334ddeg }} | |||
Badness (Sintel): 1.14 | |||
== References == | |||
== References == | |||
[[Category:Ragismic microtemperaments| ]] <!-- main article --> | |||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category: | [[Category:Catalogs of rank-2 temperaments]] | ||