Ragismic microtemperaments: Difference between revisions

Temperaments discussed elsewhere are:  

* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]

* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]

* ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments #Dzelic|37th-octave temperaments]]

 

The generator for supermajor temperament is a supermajor third, [[9/7]], tuned about 0.002 cents flat. Note that in the data that follow, the generator is given as its [[octave complement]]. 37 of these give 3/2²², 46 give 5/2²⁷, and 75 give 7/2⁴⁵. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: [[1106edo]] or [[1277edo]] can be used as tunings, leading to accuracy even greater than that of [[ennealimmal]]. The 80-note generator chain is presumably the place to start, and if that is not enough notes for you, there is always the 171-note generator chain.

{{Mapping|legend=1| 1 -22 -27 -45 | 0 37 46 75 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0067{{c}}, ~14/9 = 764.9222{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 764.9181{{c}}

: error map: {{val| 0.000 +0.013 -0.083 +0.029 }}

{{Optimal ET sequence|legend=1| 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214 }}

[[Badness]] (Sintel): 0.274

Mapping: {{mapping| 2 -7 -8 -15 -6 | 0 37 46 75 47 }}

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 164.9180{{c}}

{{Optimal ET sequence|legend=0| 80, 262d, 342, 764, 1106, 1448, 2554, 4002e, 6556cee }}

Badness (Sintel): 0.422

 

Enneadecal 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 [[6/5|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]] up to just ones.  

[[171edo]] is a good tuning for either the 5- or 7-limit, and [[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]] for a tuning.

[[Optimal tuning]]s:

* [[WE]]: ~28/27 = 63.1599{{c}}, ~3/2 = 701.9027{{c}} (~225/224 = 7.1437{{c}})

: [[error map]]: {{val| +0.038 -0.014 -0.134 +0.080 }}

* [[CWE]]: ~28/27 = 63.1579{{c}}, ~3/2 = 701.9002{{c}} (~225/224 = 7.1634{{c}})

[[Badness]] (Sintel): 0.277

Optimal tunings:

* WE: ~28/27 = 63.1431{{c}}, ~3/2 = 702.1956{{c}} (~225/224 = 7.6216{{c}})

* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.3164{{c}} (~225/224 = 7.5795{{c}})

{{Optimal ET sequence|legend=0| 19, 133d, 152, 323e, 475de, 627de }}

Badness (Sintel): 1.45

Optimal tunings:

* WE: ~28/27 = 63.1406{{c}}, ~3/2 = 702.0192{{c}} (~225/224 = 7.4730{{c}})

* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.1539{{c}} (~225/224 = 7.4171{{c}})

{{Optimal ET sequence|legend=0| 19, 133df, 152f, 323ef }}

Badness (Sintel): 1.39

Optimal tunings:

* WE: ~55/54 = 31.5800{{c}}, ~3/2 = 701.9053{{c}} (~243/242 = 7.1448{{c}})

* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9034{{c}} (~243/242 = 7.1666{{c}})

{{Optimal ET sequence|legend=0| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}

Badness (Sintel): 0.330

Optimal tunings:

* WE: ~55/54 = 31.5785{{c}}, ~3/2 = 701.9995{{c}} (~243/242 = 7.2727{{c}})

* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 702.0053{{c}} (~243/242 = 7.2685{{c}})

{{Optimal ET sequence|legend=0| 152f, 342f, 494 }}

Badness (Sintel): 0.859

Optimal tunings:

* WE: ~55/54 = 31.5784{{c}}, ~3/2 = 701.9736{{c}} (~243/242 = 7.2493{{c}})

* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9855{{c}} (~243/242 = 7.2487{{c}})

{{Optimal ET sequence|legend=0| 152, 342, 494, 1330, 1824, 2318d }}

Badness (Sintel): 1.26

Optimal tunings:  

* WE: ~55/54 = 31.5799{{c}}, ~429/250 = 935.1824{{c}} (~144/143 = 12.2152{{c}})

* CWE: ~55/54 = 31.5789{{c}}, ~429/250 = 935.1617{{c}} (~144/143 = 12.2067{{c}})

{{Optimal ET sequence|legend=0| 190, 304d, 494, 684, 1178, 2850, 4028ce }}

Badness (Sintel): 0.607

Optimal tunings:

* WE: ~28/27 = 63.1582{{c}}, ~6545/5928 = 171.2448{{c}}

* CWE: ~28/27 = 63.1579{{c}}, ~6545/5928 = 171.2439{{c}}

{{Optimal ET sequence|legend=0| 855, 988, 1843 }}

 

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimi]].''

The generator of semidimi 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.

{{Mapping|legend=1| 1 -19 -25 -32 | 0 55 73 93 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0018{{c}}, ~35/27 = 449.1277{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 449.1270{{c}}

{{Optimal ET sequence|legend=1| 8d, …, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}

[[Badness]] (Sintel): 0.382

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}}).

[[Comma list]]: 4375/4374, {{monzo| 46 -14 -3 -6 }}

[[Optimal tuning]]s:

* [[WE]]: ~1157625/1048576 = 171.4275{{c}}, ~27/20 = 519.7125{{c}}

{{Optimal ET sequence|legend=1| 7, …, 217, 224, 441, 1106, 1547 }}

[[Badness]] (Sintel): 0.737

Optimal tunings:

* WE: ~243/220 = 171.4208{{c}}, ~27/20 = 519.6807{{c}}

* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7034{{c}}

{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665 }}

Badness (Sintel): 1.73

Optimal tunings:

* WE: ~243/220 = 171.4197{{c}}, ~27/20 = 519.6789{{c}}

* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7052{{c}}

{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1106e }}

Badness (Sintel): 0.956

: ''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, 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>

{{Mapping|legend=1| 2 -4 -11 18 | 0 11 24 -19 }}

[[Optimal tuning]]s:  

* [[WE]]: ~46305/32768 = 599.9699{{c}}, ~1536/1225 = 391.0818{{c}}

: [[error map]]: {{val| -0.060 +0.065 -0.021 +0.079 }}

* [[CWE]]: ~46305/32768 = 600.0000{{c}}, ~1536/1225 = 391.1007{{c}}

: error map: {{val| 0.000 +0.152 +0.102 +0.262 }}

{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd }}

[[Badness]] (Sintel): 0.936

Mapping: {{mapping| 2 -4 -11 18 18 | 0 11 24 -19 -17 }}

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~1536/1225 = 391.0992{{c}}

{{Optimal ET sequence|legend=0| 46, 132, 178, 224, 270, 494, 764 }}

Badness (Sintel): 0.425

Mapping: {{mapping| 2 -4 -11 18 18 25 | 0 11 24 -19 -17 -27 }}

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~351/280 = 391.0969{{c}}

{{Optimal ET sequence|legend=0| 46, 178, 224, 270, 494, 764, 1258 }}

Badness (Sintel): 0.366

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Gamera]].''

{{Mapping|legend=1| 1 -17 -30 2 | 0 23 40 1 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.8483{{c}}, ~7/4 = 969.5415{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 969.6608{{c}}

[[Badness]] (Sintel): 0.953

Mapping: {{mapping| 2 -11 -20 5 10 | 0 23 40 1 -5 }}

Optimal tunings:  

* WE: ~99/70 = 599.9323{{c}}, ~99/80 = 369.6212{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~99/80 = 369.6610{{c}}

{{Optimal ET sequence|legend=0| 26, 172c, 198, 224, 422, 646, 1068d }}

Badness (Sintel): 1.35

Mapping: {{mapping| 2 -11 -20 5 10 -8 | 0 23 40 1 -5 25 }}

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~26/21 = 369.6603{{c}}

{{Optimal ET sequence|legend=0| 26, 172cf, 198, 224, 422, 646f, 1068df }}

Badness (Sintel): 0.844

Mapping: {{mapping| 1 -40 -70 1 -77 | 0 46 80 2 89 }}

Optimal tunings:  

* WE: ~2 = 1199.8845{{c}}, ~144/77 = 1084.7314{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8345{{c}}

{{Optimal ET sequence|legend=0| 73, 125, 198, 323, 521 }}

Badness (Sintel): 2.59

Mapping: {{mapping| 1 -40 -70 1 -77 -131 | 0 46 80 2 89 149 }}

Optimal tunings:

* WE: ~2 = 1199.8726{{c}}, ~144/77 = 1084.7220{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8359{{c}}

{{Optimal ET sequence|legend=0| 73f, 125f, 198, 323, 521 }}

Badness (Sintel): 1.82

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.  

* [[WE]]: ~332150625/234881024 = 600.0019{{c}}, ~1125/1024 = 162.7479{{c}}

: [[error map]]: {{val| +0.004 +0.030 -0.042 -0.014 }}

* [[CWE]]: ~332150625/234881024 = 600.0000{{c}}, ~1125/1024 = 162.7474{{c}}

: error map: {{val| 0.000 +0.024 -0.051 -0.022 }}

[[Badness]] (Sintel): 0.998

* WE: ~99/70 = 600.0047{{c}}, ~1125/1024 = 162.7493{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~1125/1024 = 162.7481{{c}}

Badness (Sintel): 0.562

[[Comma list]]: 4375/4374, {{monzo| 41 -4 2 -14 }}

{{Mapping|legend=1| 2 -8 -15 6 | 0 29 51 -1 }}

[[Optimal tuning]]s:  

* [[WE]]: ~7411887/5242880 = 599.9927{{c}}, ~8/7 = 231.1012{{c}}

: [[error map]]: {{val| -0.015 +0.037 -0.045 +0.029 }}

* [[CWE]]: ~7411887/5242880 = 600.0000{{c}}, ~8/7 = 231.1037{{c}}

{{Optimal ET sequence|legend=1| 26, …, 244, 270, 836, 1106, 1376, 2482 }}

[[Badness]] (Sintel): 1.02

Mapping: {{mapping| 2 -8 -15 6 10 | 0 29 51 -1 -8 }}

Optimal tunings:

* WE: ~99/70 = 600.0025{{c}}, ~8/7 = 231.1039{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1030{{c}}

{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836, 1106 }}

Badness (Sintel): 0.535

Mapping: {{mapping| 2 -8 -15 6 10 -3 | 0 29 51 -1 -8 27 }}

Optimal tunings:

* WE: ~99/70 = 600.0192{{c}}, ~8/7 = 231.1102{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1033{{c}}

{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836f, 1106f }}

Badness (Sintel): 0.899

: ''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.

{{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0745{{c}}, ~5120/3087 = 877.2500{{c}}

* [[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 (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.

Mapping: {{mapping| 1 -24 -43 5 2 | 0 35 62 -3 2 }}

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| 26, 119c, 145, 171, 316e }}

Badness (Sintel): 3.05

Mapping: {{mapping| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2038{{c}}

{{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }}

Badness (Sintel): 1.85

Mapping: {{mapping| 1 -24 -43 5 2 -27 -31 | 0 35 62 -3 2 42 48 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}}

{{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }}

Badness (Sintel): 1.35

: ''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.

{{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }}

: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0128{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9895{{c}}

* [[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=1| 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd }}

[[Badness]] (Sintel): 1.18

Mapping: {{mapping| 1 0 -27 109 -159 | 0 2 37 -134 205 }}

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

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 ET sequence|legend=0| 53, 559, 612 }}

Badness (Sintel): 2.22

: ''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]].

{{Mapping|legend=1| 1 -10 -13 -17 | 0 31 41 53 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9936{{c}}, ~35/27 = 448.4533{{c}}

{{Optimal ET sequence|legend=1| 8d, …, 91, 99, 289, 388, 875 }}

[[Badness]] (Sintel): 1.40

Mapping: {{mapping| 2 -20 -26 -34 -17 | 0 31 41 53 32 }}

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4546{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 190, 388 }}

Badness (Sintel): 1.95

Mapping: {{mapping| 2 -20 -26 -34 -17 -21 | 0 31 41 53 32 38 }}

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4549{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 190, 198, 388 }}

Badness (Sintel): 1.28

{{Mapping|legend=1| 1 -22 -22 -65 | 0 32 33 92 }}

[[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 }}

[[Badness]] (Sintel): 1.42

Mapping: {{mapping| 1 -22 -22 -65 58 | 0 32 33 92 -74 }}

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| 19, 251, 270, 829, 1099, 1369, 1639 }}

Badness (Sintel): 1.22

Mapping: {{mapping| 1 -22 -22 -65 58 -56 | 0 32 33 92 -74 81 }}

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| 19, 251, 270 }}

Badness (Sintel): 1.11

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

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

: ''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.  

[[Optimal tuning]]s:

* [[WE]]: ~65536/55125 = 300.0052{{c}}, ~5103/4096 = 380.3949{{c}}

[[Badness]] (Sintel): 1.56

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| 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee }}

Badness (Sintel): 0.698

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| 60d, 164, 224, 388, 612, 836 }}

Badness (Sintel): 1.22

: ''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.  

[[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 }}

[[Badness]] (Sintel): 2.04

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| 80, 190, 270, 1000, 1270, 1540e, 1810e }}

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 }}

=== 2.3.5.7.11.13.19 subgroup ===

Comma list: 1001/1000, 1521/1520, 3025/3024, 4225/4224, 4375/4374

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 }}

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.

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0068{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4467{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4466{{c}}

[[Badness]] (Sintel): 2.17

Optimal tunings:

* 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

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

: ''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.  

[[Optimal tuning]]s:

* [[CWE]]: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0030{{c}}

[[Badness]] (Sintel): 3.20

Optimal tunings:

* 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

Optimal tunings:

* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0098{{c}}

 

Badness (Sintel): 1.18

: ''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''.  

{{Mapping|legend=1| 1 -3 -15 40 | 0 9 34 -73 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.8189{{c}}, ~65536/45927 = 611.2850{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~65536/45927 = 611.3775{{c}}

: error map: {{val| 0.000 +0.443 +0.522 +0.615 }}

[[Badness]] (Sintel): 3.37

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 ET sequence|legend=0| 53, 316e, 369, 422, 791e, 1213cde }}

Badness (Sintel): 2.34

Subgroup: 2.3.5.7.11.13

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 ET sequence|legend=0| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}