Ragismic microtemperaments: Difference between revisions

This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the ragisma, [[4375/4374]] ({{monzo| -1 -7 4 1 }}). The ragisma is the smallest [[7-limit]] [[superparticular ratio]].  

Since {{nowrap|(10/9)⁴ {{=}} (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)² }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.

* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]]

* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]

* ''[[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]]

* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]

* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]

* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]

* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]

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.  

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.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 52734375/52706752

: mapping generators: ~2, ~14/9

* [[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

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 35156250/35153041

: mapping generators: ~99/70, ~11/10

Optimal tunings:  

* WE: ~99/70 = 600.0103{{c}}, ~11/10 = 164.9205{{c}}

* 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

: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Enneadecal (5-limit)]].''

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.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 703125/702464

: mapping generators: ~28/27, ~3

[[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 }}

{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}

[[Badness]] (Sintel): 0.277

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 16384/16335

Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}

* 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

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 2205/2197

Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}

Optimal tunings:  

* 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

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 234375/234256

: mapping generators: ~55/54, ~3

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

==== Hemienneadecalis ====

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256

Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}

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

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213

Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}

Optimal tunings:  

* 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

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078

Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}

: mapping generators: ~55/54, ~429/250

Optimal tunings:  

* 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

Named after the 19th element, potassium, and after an archaic variant of the element's name to resolve a name conflict. [[19/16]] can be used as a generator. Since it is enfactored in the 17-limit and lower, it makes no sense to name it for the lower subgroups.

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344

Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}

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 }}

Badness (Sintel): 3.15

: ''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.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 3955078125/3954653486

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

: mapping generators: ~2, ~35/27

[[Optimal tuning]]s:  

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

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

: error map: {{val| 0.000 +0.030 -0.043 -0.015 }}

{{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}}).

[[Subgroup]]: 2.3.5.7

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

: mapping generators: ~1157625/1048576, ~27/20

[[Optimal tuning]]s:  

: error map: {{val| 0.000 +0.049 -0.018 +0.017 }}

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

[[Badness]] (Sintel): 0.737

Subgroup: 2.3.5.7.11

Comma list: 4000/3993, 4375/4374, 131072/130977

Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}

* 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

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374

Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 2147483648/2144153025

: mapping generators: ~46305/32768, ~1536/1225

* [[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

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 131072/130977

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

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095

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

== Gamera ==

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 589824/588245

{{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}}

{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}

[[Badness]] (Sintel): 0.953

=== Hemigamera ===

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 589824/588245

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

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024

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

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 14641/14580, 15488/15435

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

* 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

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580

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

Optimal tunings:  

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

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

Badness (Sintel): 1.82

: ''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, 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

[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}

: mapping generators: ~332150625/234881024, ~1125/1024

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

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

{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}

[[Badness]] (Sintel): 0.998

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 2791309312/2790703125

Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}

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

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

{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}

Badness (Sintel): 0.562

Orga may be described as the {{nowrap| 26 & 270 }} temperament, and [[1106edo]] gives a strong tuning.

[[Subgroup]]: 2.3.5.7

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

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

: mapping generators: ~7411887/5242880, ~8/7

[[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 }}

: error map: {{val| 0.000 +0.053 -0.023 +0.070 }}

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

[[Badness]] (Sintel): 1.02

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 5767168/5764801

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

Optimal tunings:  

* 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

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360

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

Optimal tunings:  

* 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 [[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

[[Comma list]]: 4375/4374, {{monzo| -49 4 22 -3 }}

{{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }}

[[Optimal tuning]]s:  

* [[WE]]: ~25/24 = 70.5880{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9962{{c}}

* [[CWE]]: ~25/24 = 70.5882{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9990{{c}}

[[Badness]] (Sintel): 1.05

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 41503/41472, 1879453125/1879048192

Mapping: {{mapping| 17 0 26 -87 207 | 0 2 1 10 -11 }}

Optimal tunings:  

* CWE: ~25/24 = 70.5882{{c}}, ~693/400 = 950.9754{{c}}

{{Optimal ET sequence|legend=0| 289, 323, 612, 3349de, 3961de, …, 5797ddee }}

Badness (Sintel): 2.11

: {{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 generators: ~49/45, ~7/5

[[Optimal tuning]]s:  

: [[error map]]: {{val| +0.002 -0.130 -0.547 +0.883 }}

: 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]

[[Badness]] (Sintel): 1.08

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 tradeoff: ~7/5 = [582.512, 585.084]

{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224, 824d }}

Badness (Sintel): 0.466

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 }}

* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}

{{Optimal ET sequence|legend=0| 72, 152f, 224 }}

Badness (Sintel): 0.631

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 715/714, 729/728

Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}

Optimal tunings:  

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}

{{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}

Badness (Sintel): 0.729

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:  

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}}

{{Optimal ET sequence|legend=0| 72, 152fg, 224 }}

Badness (Sintel): 0.975

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 }}

Optimal tunings:  

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}

Badness (Sintel): 0.896

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 289/288, 325/324, 540/539

Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}

Optimal tunings:  

* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}

Badness (Sintel): 0.795

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=0| 8d, 72, 152 }}

Badness (Sintel): 0.993

Scales: [[Octoid72]], [[Octoid80]]

{{See also| 16th-octave temperaments }}

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}}

{{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

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444

Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}

Optimal tunings:  

* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}

{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}

Badness (Sintel): 1.44

: ''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

[[Comma list]]: 4375/4374, 201768035/201326592

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

: mapping generators: ~2, ~5120/3087

[[Optimal tuning]]s:  

* [[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.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4374, 65536/65219

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

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 2200/2197, 4375/4374

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

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| 26, 119cf, 145, 171, 316ef }}

Badness (Sintel): 1.85

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

: ''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

[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}

{{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}}

: 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

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 41503/41472, 184549376/184528125

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

Optimal tunings: