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Ragismic microtemperaments

Revision as of 15:00, 27 March 2026 by FloraC (talk | contribs) (Switch to Sintel's badness, WE & CWE tunings (2/))
This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-2 temperaments tempering out the ragisma, 4375/4374 ([-1 -7 4 1⟩). The ragisma is the smallest 7-limit superparticular ratio.

Since (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 microtemperaments 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 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.

Microtemperaments considered below, sorted by badness, are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:

Supermajor

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: [⟨1 -22 -27 -45], ⟨0 37 46 75]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1200.0067 ¢, ~14/9 = 764.9222 ¢
error map: ⟨+0.007 +0.019 -0.074 +0.037]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 764.9181 ¢
error map: ⟨0.000 +0.013 -0.083 +0.029]

Optimal ET sequence: 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214

Badness (Sintel): 0.274

Semisupermajor

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 -7 -8 -15 -6], ⟨0 37 46 75 47]]

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

Optimal tunings:

  • WE: ~99/70 = 600.0103 ¢, ~11/10 = 164.9205 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~11/10 = 164.9180 ¢

Optimal ET sequence: 80, 262d, 342, 764, 1106, 1448, 2554, 4002e, 6556cee

Badness (Sintel): 0.422

Enneadecal

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

Enneadecal tempers out the enneadeca, [-14 -19 19⟩, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)1/3. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of 19edo 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: [⟨19 0 14 -37], ⟨0 1 1 3]]

mapping generators: ~28/27, ~3

Optimal tunings:

  • WE: ~28/27 = 63.1599 ¢, ~3/2 = 701.9027 ¢ (~225/224 = 7.1437 ¢)
error map: ⟨+0.038 -0.014 -0.134 +0.080]
  • CWE: ~28/27 = 63.1579 ¢, ~3/2 = 701.9002 ¢ (~225/224 = 7.1634 ¢)
error map: ⟨0.000 -0.055 -0.203 +0.033]

Optimal ET sequence: 19, …, 152, 171, 665, 836, 1007, 2185, 3192c

Badness (Sintel): 0.277

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨19 0 14 -37 126], ⟨0 1 1 3 -2]]

Optimal tunings:

  • WE: ~28/27 = 63.1431 ¢, ~3/2 = 702.1956 ¢ (~225/224 = 7.6216 ¢)
  • CWE: ~28/27 = 63.1579 ¢, ~3/2 = 702.3164 ¢ (~225/224 = 7.5795 ¢)

Optimal ET sequence: 19, 133d, 152, 323e, 475de, 627de

Badness (Sintel): 1.45

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨19 0 14 -37 126 -20], ⟨0 1 1 3 -2 3]]

Optimal tunings:

  • WE: ~28/27 = 63.1406 ¢, ~3/2 = 702.0192 ¢ (~225/224 = 7.4730 ¢)
  • CWE: ~28/27 = 63.1579 ¢, ~3/2 = 702.1539 ¢ (~225/224 = 7.4171 ¢)

Optimal ET sequence: 19, 133df, 152f, 323ef

Badness (Sintel): 1.39

Hemienneadecal

Subgroup: 2.3.5.7.11

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

Mapping: [⟨38 0 28 -74 11], ⟨0 1 1 3 2]]

mapping generators: ~55/54, ~3

Optimal tunings:

  • WE: ~55/54 = 31.5800 ¢, ~3/2 = 701.9053 ¢ (~243/242 = 7.1448 ¢)
  • CWE: ~55/54 = 31.5789 ¢, ~3/2 = 701.9034 ¢ (~243/242 = 7.1666 ¢)

Optimal ET sequence: 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: [⟨38 0 28 -74 11 -281], ⟨0 1 1 3 2 7]]

Optimal tunings:

  • WE: ~55/54 = 31.5785 ¢, ~3/2 = 701.9995 ¢ (~243/242 = 7.2727 ¢)
  • CWE: ~55/54 = 31.5789 ¢, ~3/2 = 702.0053 ¢ (~243/242 = 7.2685 ¢)

Optimal ET sequence: 152f, 342f, 494

Badness (Sintel): 0.859

Hemienneadec

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨38 0 28 -74 11 502], ⟨0 1 1 3 2 -6]]

Optimal tunings:

  • WE: ~55/54 = 31.5784 ¢, ~3/2 = 701.9736 ¢ (~243/242 = 7.2493 ¢)
  • CWE: ~55/54 = 31.5789 ¢, ~3/2 = 701.9855 ¢ (~243/242 = 7.2487 ¢)

Optimal ET sequence: 152, 342, 494, 1330, 1824, 2318d

Badness (Sintel): 1.26

Semihemienneadecal

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨38 1 29 -71 13 111], ⟨0 2 2 6 4 1]]

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

Optimal tunings:

  • WE: ~55/54 = 31.5799 ¢, ~429/250 = 935.1824 ¢ (~144/143 = 12.2152 ¢)
  • CWE: ~55/54 = 31.5789 ¢, ~429/250 = 935.1617 ¢ (~144/143 = 12.2067 ¢)

Optimal ET sequence: 190, 304d, 494, 684, 1178, 2850, 4028ce

Badness (Sintel): 0.607

Kalium

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: [⟨19 3 17 -28 82 92 159 78], ⟨0 10 10 30 -6 -8 -30 1]]

Optimal tunings:

  • WE: ~28/27 = 63.1582 ¢, ~6545/5928 = 171.2448 ¢
  • CWE: ~28/27 = 63.1579 ¢, ~6545/5928 = 171.2439 ¢

Optimal ET sequence: 855, 988, 1843

Badness (Sintel): 3.15

Semidimi

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 [-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: [⟨1 -19 -25 -32], ⟨0 55 73 93]]

mapping generators: ~2, ~35/27

Optimal tunings:

  • WE: ~2 = 1200.0018 ¢, ~35/27 = 449.1277 ¢
error map: ⟨+0.002 +0.031 -0.040 -0.012]
  • CWE: ~2 = 1200.0000 ¢, ~35/27 = 449.1270 ¢
error map: ⟨0.000 +0.030 -0.043 -0.015]

Optimal ET sequence: 8d, …, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419

Badness (Sintel): 0.382

Brahmagupta

The brahmagupta temperament has a period of 1/7 octave, tempering out the akjaysma ([47 -7 -7 -7⟩).

Early in the design of the Sagittal notation system, Secor and 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 ¢ many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of Brahmagupta temperament 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 ¢) and 1/21-apotome generator (5.4136 ¢).

Subgroup: 2.3.5.7

Comma list: 4375/4374, 70368744177664/70338939985125

Mapping: [⟨7 2 -8 53], ⟨0 3 8 -11]]

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

Optimal tuning (POTE): ~1157625/1048576 = 171.429 ¢, ~27/20 = 519.716 ¢

Optimal ET sequence: 217, 224, 441, 1106, 1547

Badness (Sintel): 0.737

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨7 2 -8 53 3], ⟨0 3 8 -11 7]]

Optimal tuning (POTE): ~243/220 = 171.429 ¢, ~27/20 = 519.704 ¢

Optimal ET sequence: 7, 217, 224, 441, 665, 1771ee

Badness (Sintel): 1.725

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨7 2 -8 53 3 35], ⟨0 3 8 -11 7 -3]]

Optimal tuning (POTE): ~243/220 = 171.429 ¢, ~27/20 = 519.706 ¢

Optimal ET sequence: 7, 217, 224, 441, 665, 1771eef

Badness (Sintel): 0.956

Abigail

For the 5-limit versino, see Miscellaneous 5-limit temperaments #Abigail.

Abigail temperament tempers out the pessoalisma in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.[1]

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2147483648/2144153025

Mapping: [⟨2 7 13 -1], ⟨0 -11 -24 19]]

Mapping generators: ~46305/32768, ~27/20

Optimal tuning (POTE): ~46305/32768 = 600.000 ¢, ~6912/6125 = 208.899 ¢

Optimal ET sequence: 46, 132, 178, 224, 270, 494, 764, 1034, 1798

Badness (Sintel): 0.936

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 7 13 -1 1], ⟨0 -11 -24 19 17]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~1155/1024 = 208.901 ¢

Optimal ET sequence: 46, 132, 178, 224, 270, 494, 764

Badness (Sintel): 0.425

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 7 13 -1 1 -2], ⟨0 -11 -24 19 17 27]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~44/39 = 208.903 ¢

Optimal ET sequence: 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: [⟨1 6 10 3], ⟨0 -23 -40 -1]]

mapping generators: ~2, ~8/7

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~8/7 = 230.336 ¢

Optimal ET sequence: 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: [⟨2 12 20 6 5], ⟨0 -23 -40 -1 5]]

mapping generators: ~99/70, ~8/7

Optimal tuning (POTE): ~99/70 = 600.0000 ¢, ~8/7 = 230.3370 ¢

Optimal ET sequence: 26, 198, 224, 422, 646, 1068d

Badness (Sintel): 1.354

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 12 20 6 5 17], ⟨0 -23 -40 -1 5 -25]]

Optimal tuning (POTE): ~99/70 = 600.0000 ¢, ~8/7 = 230.3373 ¢

Optimal ET sequence: 26, 198, 224, 422, 646f, 1068df

Badness (Sintel): 0.844

Semigamera

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 6 10 3 12], ⟨0 -46 -80 -2 -89]]

mapping generators: ~2, ~77/72

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~77/72 = 115.1642 ¢

Optimal ET sequence: 73, 125, 198, 323, 521

Badness (Sintel): 2.589

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 6 10 3 12 18], ⟨0 -46 -80 -2 -89 -149]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~77/72 = 115.1628 ¢

Optimal ET sequence: 73f, 125f, 198, 323, 521

Badness (Sintel): 1.821

Crazy

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 118 & 494 temperament. 1106edo is a strong tuning.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-53 10 16⟩

Mapping: [⟨2 1 6 -15], ⟨0 8 -5 76]]

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

Optimal tunings:

  • CTE: ~332150625/234881024 = 600.0000 ¢, ~1125/1024 = 162.7475 ¢
  • error map: ⟨0.0000 +0.0253 -0.0514 -0.0133]
  • CWE: ~332150625/234881024 = 600.0000 ¢, ~1125/1024 = 162.7474 ¢
  • error map: ⟨0.0000 +0.0244 -0.0508 -0.0218]

Optimal ET sequence: 118, 376, 494, 612, 1106, 1718

Badness (Sintel): 0.998

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 1 6 -15 -8], ⟨0 8 -5 76 55]]

Optimal tunings:

  • CTE: ~99/70 = 162.7485 ¢, ~1125/1024 = 162.7485 ¢
  • CWE: ~99/70 = 162.7485 ¢, ~1125/1024 = 162.7481 ¢

Optimal ET sequence: 118, 376, 494, 612, 1106, 2824, 3930e

Badness (Sintel): 0.562

Orga

Subgroup: 2.3.5.7

Comma list: 4375/4374, 54975581388800/54936068900769

Mapping: [⟨2 21 36 5], ⟨0 -29 -51 1]]

Mapping generators: ~7411887/5242880, ~1310720/1058841

Optimal tuning (POTE): ~7411887/5242880 = 600.000 ¢, ~8/7 = 231.104 ¢

Optimal ET sequence: 26, 244, 270, 836, 1106, 1376, 2482

Badness (Sintel): 1.018

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 21 36 5 2], ⟨0 -29 -51 1 8]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~8/7 = 231.103 ¢

Optimal ET sequence: 26, 244, 270, 566, 836, 1106

Badness (Sintel): 0.535

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 21 36 5 2 24], ⟨0 -29 -51 1 8 -27]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~8/7 = 231.103 ¢

Optimal ET sequence: 26, 244, 270, 566, 836f, 1106f

Badness (Sintel): 0.899

Seniority

For the 5-limit version, see Very high accuracy temperaments #Senior.

Aside from the ragisma, the seniority temperament (26 & 145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ([-17 62 -35⟩, quadla-sepquingu) is tempered out.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 201768035/201326592

Mapping: [⟨1 11 19 2], ⟨0 -35 -62 3]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3087/2560 = 322.804 ¢

Optimal ET sequence: 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d

Badness (Sintel): 1.136

Senator

The senator temperament (26 & 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It 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: [⟨1 11 19 2 4], ⟨0 -35 -62 3 -2]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~77/64 = 322.793 ¢

Optimal ET sequence: 26, 119c, 145, 171, 316e, 487ee

Badness (Sintel): 3.049

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 11 19 2 4 15], ⟨0 -35 -62 3 -2 -42]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~77/64 = 322.793 ¢

Optimal ET sequence: 26, 119c, 145, 171, 316ef, 487eef

Badness (Sintel): 1.845

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197

Mapping: [⟨1 11 19 2 4 15 17], ⟨0 -35 -62 3 -2 -42 -48]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~77/64 = 322.793 ¢

Optimal ET sequence: 26, 119c, 145, 171, 316ef, 487eef

Badness (Sintel): 1.353

Monzismic

For the 5-limit version, see Very high accuracy temperaments #Monzismic.

The monzismic temperament (53 & 612) tempers out the monzisma, [54 -37 2⟩, and in the 7-limit, the nanisma, [109 -67 0 -1⟩, as well as the ragisma, 4375/4374. 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, [-55 30 2 1⟩

Mapping: [⟨1 2 10 -25], ⟨0 -2 -37 134]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~[-27 11 3 1⟩ = 249.0207 ¢

Optimal ET sequence: 53, …, 559, 612, 1277, 1889

Badness (Sintel): 1.179

Monzism

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 10 -25 46], ⟨0 -2 -37 134 -205]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~231/200 = 249.0193 ¢

Optimal ET sequence: 53, 559, 612

Badness (Sintel): 1.887

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625

Mapping: [⟨1 2 10 -25 46 23], ⟨0 -2 -37 134 -205 -93]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~231/200 = 249.0199 ¢

Optimal ET sequence: 53, 559, 612

Badness (Sintel): 2.222

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

Comma list: 4375/4374, 235298/234375

Mapping: [⟨1 21 28 36], ⟨0 -31 -41 -53]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~35/27 = 448.456 ¢

Optimal ET sequence: 8d, 91, 99, 289, 388, 875, 1263d, 1651d

Badness (Sintel): 1.398

Neusec

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 235298/234375

Mapping: [⟨2 11 15 19 15], ⟨0 -31 -41 -53 -32]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~12/11 = 151.547 ¢

Optimal ET sequence: 8d, 190, 388

Badness (Sintel): 1.955

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374

Mapping: [⟨2 11 15 19 15 17], ⟨0 -31 -41 -53 -32 -38]]

Optimal tuning (POTE): ~99/70 = 1200.000 ¢, ~12/11 = 151.545 ¢

Optimal ET sequence: 8d, 190, 198, 388

Badness (Sintel): 1.279

Acrokleismic

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2202927104/2197265625

Mapping: [⟨1 10 11 27], ⟨0 -32 -33 -92]]

mapping generators: ~2, ~6/5

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 315.557 ¢

Optimal ET sequence: 19, …, 251, 270, 2449c, 2719c, 2989bc

Badness (Sintel): 1.422

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 41503/41472, 172032/171875

Mapping: [⟨1 10 11 27 -16], ⟨0 -32 -33 -92 74]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 315.558 ¢

Optimal ET sequence: 19, 251, 270, 829, 1099, 1369, 1639

Badness (Sintel): 1.219

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976

Mapping: [⟨1 10 11 27 -16 25], ⟨0 -32 -33 -92 74 -81]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 315.557 ¢

Optimal ET sequence: 19, 251, 270

Badness (Sintel): 1.108

Counteracro

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 117649/117612

Mapping: [⟨1 10 11 27 55], ⟨0 -32 -33 -92 -196]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 315.553 ¢

Optimal ET sequence: 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde

Badness (Sintel): 1.407

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374

Mapping: [⟨1 10 11 27 55 25], ⟨0 -32 -33 -92 -196 -81]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 315.554 ¢

Optimal ET sequence: 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf

Badness (Sintel): 1.076

Quasithird

The quasithird temperament is featured by a major third interval which is 1600000/1594323 (amity comma) or 5120/5103 (hemifamity comma) below the just major third 5/4 as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the ragisma and [-60 29 0 5⟩.

Subgroup: 2.3.5

Comma list: [55 -64 20⟩

Mapping: [⟨4 0 -11], ⟨0 5 16]]

mapping generators: ~51200000/43046721, ~1594323/1280000

Optimal tuning (POTE): ~51200000/43046721 = 300.000 ¢, ~1594323/1280000 = 380.395 ¢

Optimal ET sequence: 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404

Badness (Sintel): 2.335

7-limit

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-60 29 0 5⟩

Mapping: [⟨4 0 -11 48], ⟨0 5 16 -29]]

Optimal tuning (POTE): ~65536/55125 = 300.000 ¢, ~5103/4096 = 380.388  ¢

Optimal ET sequence: 60d, 164, 224, 388, 612, 1448, 2060

Badness (Sintel): 1.564

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 4296700485/4294967296

Mapping: [⟨4 0 -11 48 43], ⟨0 5 16 -29 -23]]

Optimal tuning (POTE): ~65536/51125 = 300.000 ¢, ~5103/4096 = 380.387 ¢ (or ~22/21 = 80.387 ¢)

Optimal ET sequence: 60d, 164, 224, 388, 612, 836, 1448

Badness (Sintel): 0.698

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374

Mapping: [⟨4 0 -11 48 43 11], ⟨0 5 16 -29 -23 3]]

Optimal tuning (POTE): ~65536/51125 = 300.000 ¢, ~81/65 = 380.385 ¢ (or ~22/21 = 80.385 ¢)

Optimal ET sequence: 60d, 164, 224, 388, 612, 836, 1448f, 2284f

Badness (Sintel): 1.219

Deca

For 5-limit version, see 10th-octave temperaments #Neon.

Deca temperament has a period of 1/10 octave and tempers out the linus comma, [11 -10 -10 10⟩, neon comma [21 60 -50⟩ and [12 -3 -14 9⟩ = 165288374272/164794921875 (satritrizo-asepbigu).

Subgroup: 2.3.5.7

Comma list: 4375/4374, 165288374272/164794921875

Mapping: [⟨10 4 9 2], ⟨0 5 6 11]]

mapping generators: ~15/14, ~6/5

Optimal tuning (POTE): ~15/14 = 120.000 ¢, ~6/5 = 315.577 ¢

Optimal ET sequence: 80, 190, 270, 1270, 1540, 1810, 2080

Badness (Sintel): 2.041

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 391314/390625

Mapping: [⟨10 4 9 2 18], ⟨0 5 6 11 7]]

Optimal tuning (POTE): ~15/14 = 120.000 ¢, ~6/5 = 315.582 ¢

Optimal ET sequence: 80, 190, 270, 1000, 1270, 1540e, 1810e

Badness (Sintel): 0.804

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨10 4 9 2 18 37], ⟨0 5 6 11 7 0]]

Optimal tuning (POTE): ~15/14 = 120.000 ¢, ~6/5 = 315.602 ¢ (~40/39 = 44.398 ¢)

Optimal ET sequence: 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, 3025/3024, 4225/4224, 4375/4374, 1521/1520

Mapping: [⟨10 4 9 2 18 37 33], ⟨0 5 6 11 7 0 4]]

Optimal tuning (CTE): ~15/14 = 120.000 ¢, ~6/5 = 315.581 ¢ (~39/38 = 44.419 ¢)

Optimal ET sequence: 80, 190, 270, 730, 1000

Badness (Sintel): 0.556

Keenanose

Keenanose is named for the fact that it uses 385/384, the keenanisma, as the generator.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-56 1 -8 26⟩

Mapping: [⟨1 2 3 3], ⟨0 -112 -183 -52]]

mapping generators: ~2, ~[21 3 1 -10⟩

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~[21 3 1 -10⟩ = 4.4465 ¢

Optimal ET sequence: 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd

Badness (Sintel): 2.172

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 117649/117612, 67110351/67108864

Mapping: [⟨1 2 3 3 3], ⟨0 -112 -183 -52 124]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~385/384 = 4.4465 ¢

Optimal ET sequence: 270, 1349, 1619, 1889, 2159, 11065, 13224

Badness (Sintel): 1.020

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612

Mapping: [⟨1 2 3 3 3 3], ⟨0 -112 -183 -52 124 189]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~385/384 = 4.4466 ¢

Optimal ET sequence: 270, 1079, 1349, 1619, 1889, 4048

Badness (Sintel): 0.879

Aluminium

For the 5-limit version, see 13th-octave temperaments #Aluminium.

Aluminium is named after the 13th element, and tempers out the [92 -39 -13⟩ comma which sets 135/128 interval to be equal to 1/13th of the octave.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [92 -39 -13⟩

Mapping: [⟨13 0 92 -355], ⟨0 1 -3 19]]

Mapping generators: ~135/128, ~3

Optimal tuning (CTE): ~135/128 = 92.3077 ¢, ~3/2 = 702.0024 ¢

Optimal ET sequence: 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b

Badness (Sintel): 3.201

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 234375/234256, 2097152/2096325

Mapping: [⟨13 0 92 -355 148], ⟨0 1 -3 19 -5]]

Optimal tuning (CTE): ~135/128 = 92.3077 ¢, ~3/2 = 702.0042 ¢

Optimal ET sequence: 494, 1053, 1547, 3588e, 5135e

Badness (Sintel): 1.393

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078

Mapping: [⟨13 0 92 -355 148 419], ⟨0 1 -3 19 -5 -18]]

Optimal tuning (CTE): ~135/128 = 92.3077 ¢, ~3/2 = 702.0099 ¢

Optimal ET sequence: 494, 1547, 2041, 4576def

Badness (Sintel): 1.180

Countritonic

For the 5-limit version, see Schismic–Mercator equivalence continuum #Countritonic (5-limit).

Countritonic (co-un-tritonic) can be described as the 53 & 422 temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 68719476736/68356598625

Mapping: [⟨1 6 19 -33], ⟨0 -9 -34 73]]

mapping generators: ~2, ~45927/32768

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~45927/32768 = 588.6216 ¢

Optimal ET sequence: 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd

Badness (Sintel): 3.370

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 2621440/2614689

Mapping: [⟨1 6 19 -13 79], ⟨0 -9 -34 73 154]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~539/384 = 588.6258 ¢

Optimal ET sequence: 53, 316e, 369, 422, 791e, 1213cde

Badness (Sintel): 2.336

13-limit

Subgroup: 2.3.5.7.11

Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625

Mapping: [⟨1 6 19 -13 79], ⟨0 -9 -34 73 154 -74]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~128/91 = 588.6277 ¢

Optimal ET sequence: 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff

Badness (Sintel): 1.514

Quatracot

See also: Stratosphere

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-32 5 14 -3⟩

Mapping: [⟨2 7 7 23], ⟨0 -13 -8 -59]]

mapping generators: ~2278125/1605632, ~448/405

Optimal tuning (POTE): ~2278125/1605632 = 600.000 ¢, ~448/405 = 176.805 ¢

Optimal ET sequence: 190, 224, 414, 638, 1052c, 1690bcc

Badness (Sintel): 4.454

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 1265625/1261568

Mapping: [⟨2 7 7 23 19], ⟨0 -13 -8 -59 -41]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~448/405 = 176.806 ¢

Optimal ET sequence: 190, 224, 414, 638, 1052c

Badness (Sintel): 1.357

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1575/1573, 2200/2197

Mapping: [⟨2 7 7 23 19 13], ⟨0 -13 -8 -59 -41 -19]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~195/176 = 176.804 ¢

Optimal ET sequence: 190, 224, 414, 638, 1690bcc, 2328bccde

Badness (Sintel): 0.936

Moulin

Moulin has a generator of 22/13, and it is named after the Law & Order: Special Victims Unit episode Season 22, Episode 13. "Trick-Rolled At The Moulin". It can be described as the 494 & 1619 temperament. Since 11/8 is within 23 generators, the 25-tone mos (4L 21s) of this temperament contains the 8:11:13 triad.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-88 2 45 -7⟩

Mapping: [⟨1 57 38 248], ⟨0 -73 -47 -323]]

Mapping generators: ~2, ~6422528/3796875

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~6422528/3796875 = 910.9323 ¢

Optimal ET sequence: 494, 1125, 1619

Badness (Sintel): 5.931

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 759375/758912, 100663296/100656875

Mapping: [⟨1 57 38 248 -14], ⟨0 -73 -47 -323 23]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~1024/605 = 910.9323 ¢

Optimal ET sequence: 494, 1125, 1619, 2113

Badness (Sintel): 2.240

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078

Mapping: [⟨1 57 38 248 -14 -13], ⟨0 -73 -47 -323 23 22]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~22/13 = 910.9323 ¢

Optimal ET sequence: 494, 1125, 1619, 2113

Badness (Sintel): 1.118

Palladium

For the 5-limit version of this temperament, 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, [-39 92 -46⟩, by which 46 minor whole tones (10/9) fall short of seven octaves. This temperament can be described as 46 & 414 temperament, which tempers out [-51 8 2 12⟩ as well as the ragisma.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-51 8 2 12⟩

Mapping: [⟨46 0 -39 202], ⟨0 1 2 -1]]

Mapping generators: ~83349/81920, ~3

Optimal tuning (POTE): ~83349/81920 = 26.0870 ¢, ~3/2 = 701.6074 ¢

Optimal ET sequence: 46, 368, 414, 460, 874d

Badness (Sintel): 7.807

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 134775333/134217728

Mapping: [⟨46 0 -39 202 232], ⟨0 1 2 -1 -1]]

Optimal tuning (POTE): ~8192/8085 = 26.0870 ¢, ~3/2 = 701.5951 ¢

Optimal ET sequence: 46, 368, 414, 460, 874de

Badness (Sintel): 2.439

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364

Mapping: [⟨46 0 -39 202 232 316], ⟨0 1 2 -1 -1 -2]]

Optimal tuning (POTE): ~65/64 = 26.0870 ¢, ~3/2 = 701.6419 ¢

Optimal ET sequence: 46, 368, 414, 460, 874de, 1334de

Badness (Sintel): 1.684

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224

Mapping: [⟨46 0 -39 202 232 316 188], ⟨0 1 2 -1 -1 -2 0]]

Optimal tuning (POTE): ~65/64 = 26.0870 ¢, ~3/2 = 701.6425 ¢

Optimal ET sequence: 46, 368, 414, 460, 874de, 1334deg

Badness (Sintel): 1.143

Oviminor

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Oviminor (5-limit).

Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past egads, though it is less accurate.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-100 53 48 -34⟩

Mapping: [⟨1 50 51 147], ⟨0 -184 -185 -548]]

Mapping generators: ~2, ~6/5

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~6/5 = 315.7501 ¢

Optimal ET sequence: 19, …, 1600, 1619, 4838, 6457c

Badness (Sintel): 14.739

Octoid

For the 5-limit version, see 8th-octave temperaments #Octoid.

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 16875/16807

Mapping: [⟨8 1 3 3], ⟨0 3 4 5]]

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

Optimal tuning (POTE): ~49/45 = 150.000 ¢, ~7/5 = 583.940 ¢

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: 8d, 72, 152, 224

Badness (Sintel): 1.080

Scales: octoid72, octoid80

11-limit

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 minimaxing the 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, if one wants to use 80edo as the tuning, one must use octopus – not octoid – as 80edo does not temper 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 4000/3993

Mapping: [⟨8 1 3 3 16], ⟨0 3 4 5 3]]

Optimal tuning (POTE): ~12/11 = 150.000 ¢, ~7/5 = 583.962 ¢

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: 72, 152, 224

Badness (Sintel): 0.466

Scales: octoid72, octoid80

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 1375/1372

Mapping: [⟨8 1 3 3 16 -21], ⟨0 3 4 5 3 13]]

Optimal tuning (POTE): ~12/11 = 150.000 ¢, ~7/5 = 583.905 ¢

Optimal ET sequence: 72, 152f, 224

Badness (Sintel): 0.631

Scales: octoid72, octoid80

Music
17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨8 1 3 3 16 -21 -14], ⟨0 3 4 5 3 13 12]]

Optimal tuning (POTE): ~12/11 = 150.000 ¢, ~7/5 = 583.842 ¢

Optimal ET sequence: 72, 152fg, 224, 296, 520g

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: [⟨8 1 3 3 16 -21 -14 34], ⟨0 3 4 5 3 13 12 0]]

Optimal tuning (POTE): ~12/11 = 150.000 ¢, ~7/5 = 583.932 ¢

Optimal ET sequence: 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 ¢.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 540/539

Mapping: [⟨8 1 3 3 16 14], ⟨0 3 4 5 3 4]]

Optimal tuning (POTE): ~12/11 = 150.000 ¢, ~7/5 = 583.892 ¢

Optimal ET sequence: 72, 152, 224f

Badness (Sintel): 0.0896

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨8 1 3 3 16 14 21], ⟨0 3 4 5 3 4 3]]

Optimal tuning (POTE): ~12/11 = 150.000 ¢, ~7/5 = 583.811 ¢

Optimal ET sequence: 72, 152, 224fg, 296ffg

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: [⟨8 1 3 3 16 14 21 34], ⟨0 3 4 5 3 4 3 0]]

Optimal tuning (POTE): ~12/11 = 150.000 ¢, ~7/5 = 584.064 ¢

Optimal ET sequence: 72, 152, 224fg, 376ffgh

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: [⟨16 2 6 6 32 67], ⟨0 3 4 5 3 -1]]

mapping generators: ~448/429, ~7/5

Optimal tuning (POTE): ~448/429 = 75.000 ¢, ~13/8 = 841.015 ¢

Optimal ET sequence: 80, 144, 224

Badness (Sintel): 1.273

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224

Mapping: [⟨16 2 6 6 32 67 81], ⟨0 3 4 5 3 -1 -2]]

Optimal tuning (POTE): ~117/112 = 75.000 ¢, ~13/8 = 840.932 ¢

Optimal ET sequence: 80, 144, 224, 528dg

Badness (Sintel): 1.458

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

Mapping: [⟨16 2 6 6 32 67 81 68], ⟨0 -3 -4 -5 -3 1 2 0]]

Optimal tuning (POTE): ~117/112 = 75.000 ¢, ~13/8 = 840.896 ¢

Optimal ET sequence: 80, 144, 224, 304dh, 528dghh

Badness (Sintel): 1.443

Parakleismic

Main article: Parakleismic
For the 5-limit version, see Syntonic–kleismic equivalence continuum #Parakleismic (5-limit).

In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, [8 14 -13⟩, with the 118edo tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, and 11-limit adding 385/384. For the 7-limit 99edo may be preferred, but in the 11-limit it is best to stick with 118.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 4375/4374

Mapping: [⟨1 5 6 12], ⟨0 -13 -14 -35]]

mapping generators: ~2, ~6/5

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 315.181 ¢

Optimal ET sequence: 19, 80, 99, 217, 316, 415

Badness (Sintel): 0.694

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3136/3125, 4375/4374

Mapping: [⟨1 5 6 12 -6], ⟨0 -13 -14 -35 36]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 315.251 ¢

Optimal ET sequence: 19, 99, 118

Badness (Sintel): 1.643

Paralytic

The paralytic temperament (118 & 217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 4375/4374

Mapping: [⟨1 5 6 12 25], ⟨0 -13 -14 -35 -82]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 315.220 ¢

Optimal ET sequence: 19e, 99e, 118, 217, 335, 552d, 887dd

Badness (Sintel): 1.191

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374

Mapping: [⟨1 5 6 12 25 -16], ⟨0 -13 -14 -35 -82 75]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 315.214 ¢

Optimal ET sequence: 99e, 118, 217, 552d, 769de

Badness (Sintel): 1.847

Paraklein

The paraklein temperament (19e & 118) is another 13-limit extension of paralytic, which equates 13/11 with 32/27, 14/13 with 15/14, 25/24 with 26/25, and 27/26 with 28/27.

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 625/624, 729/728

Mapping: [⟨1 5 6 12 25 15], ⟨0 -13 -14 -35 -82 -43]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 315.225 ¢

Optimal ET sequence: 19e, 99ef, 118, 217ff, 335ff

Badness (Sintel): 1.554

Parkleismic

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1375/1372, 2200/2187

Mapping: [⟨1 5 6 12 20], ⟨0 -13 -14 -35 -63]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 315.060 ¢

Optimal ET sequence: 19e, 80, 179, 259cd

Badness (Sintel): 1.848

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 1375/1372

Mapping: [⟨1 5 6 12 20 10], ⟨0 -13 -14 -35 -63 -24]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 315.075 ¢

Optimal ET sequence: 19e, 80, 179

Badness (Sintel): 1.511

Paradigmic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 3136/3125

Mapping: [⟨1 5 6 12 -1], ⟨0 -13 -14 -35 17]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 315.096 ¢

Optimal ET sequence: 19, 61d, 80, 99e, 179e

Badness (Sintel): 1.379

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 540/539, 832/825

Mapping: [⟨1 5 6 12 -1 10], ⟨0 -13 -14 -35 17 -24]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 315.080 ¢

Optimal ET sequence: 19, 61d, 80, 99e, 179e

Badness (Sintel): 1.479

Semiparakleismic

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 4375/4374

Mapping: [⟨2 10 12 24 19], ⟨0 -13 -14 -35 -23]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~6/5 = 315.181 ¢

Optimal ET sequence: 80, 118, 198, 316, 514c, 830c

Badness (Sintel): 1.131

Semiparamint

This extension was named semiparakleismic in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374

Mapping: [⟨2 10 12 24 19 -1], ⟨0 -13 -14 -35 -23 16]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~6/5 = 315.156 ¢

Optimal ET sequence: 80, 118, 198

Badness (Sintel): 1.396

Semiparawolf

This extension was named gentsemiparakleismic in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 3136/3125

Mapping: [⟨2 10 12 24 19 20], ⟨0 -13 -14 -35 -23 -24]]

Optimal tuning (POTE): ~55/39 = 600.000 ¢, ~6/5 = 315.184 ¢

Optimal ET sequence: 80, 118f, 198f

Badness (Sintel): 1.672

Counterkleismic

For the 5-limit temperament, see Syntonic–kleismic equivalence continuum #Counterhanson.

In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, [-20 -24 25⟩, the amount by which six major dieses (648/625) fall short of the classic major third (5/4). It can be described as 19 & 224 temperament (counterkleismic, named by analogy to catakleismic and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).

Subgroup: 2.3.5.7

Comma list: 4375/4374, 158203125/157351936

Mapping: [⟨1 20 20 61], ⟨0 -25 -24 -79]]

Mapping generators: ~2, ~5/3

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 316.060 ¢

Optimal ET sequence: 19, 205, 224, 243, 467

Badness (Sintel): 2.292

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 2097152/2096325

Mapping: [⟨1 20 20 61 -40], ⟨0 -25 -24 -79 59]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 316.071 ¢

Optimal ET sequence: 19, 205, 224

Badness (Sintel): 2.346

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 10985/10976

Mapping: [⟨1 20 20 61 -40 56], ⟨0 -25 -24 -79 59 -71]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 316.070 ¢

Optimal ET sequence: 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef

Badness (Sintel): 1.400

Counterlytic

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4375/4374, 496125/495616

Mapping: [⟨1 20 20 61 125], ⟨0 -25 -24 -79 -165]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 316.065 ¢

Optimal ET sequence: 19e, 205e, 224

Badness (Sintel): 2.162

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1375/1372, 10985/10976

Mapping: [⟨1 20 20 61 125 56], ⟨0 -25 -24 -79 -165 -71]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 316.065 ¢

Optimal ET sequence: 19e, 205e, 224

Badness (Sintel): 1.231

Quincy

Subgroup: 2.3.5.7

Comma list: 4375/4374, 823543/819200

Mapping: [⟨1 2 3 3], ⟨0 -30 -49 -14]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~1728/1715 = 16.613 ¢

Optimal ET sequence: 72, 217, 289

Badness (Sintel): 2.016

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4000/3993, 4375/4374

Mapping: [⟨1 2 3 3 4], ⟨0 -30 -49 -14 -39]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~100/99 = 16.613 ¢

Optimal ET sequence: 72, 217, 289

Badness (Sintel): 1.021

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675, 4375/4374

Mapping: [⟨1 2 3 3 4 5], ⟨0 -30 -49 -14 -39 -94]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~100/99 = 16.602 ¢

Optimal ET sequence: 72, 145, 217, 289

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: [⟨1 2 3 3 4 5 5], ⟨0 -30 -49 -14 -39 -94 -66]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~100/99 = 16.602 ¢

Optimal ET sequence: 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: [⟨1 2 3 3 4 5 5 4], ⟨0 -30 -49 -14 -39 -94 -66 18]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~100/99 = 16.594 ¢

Optimal ET sequence: 72, 145, 217

Badness (Sintel): 0.924

Sfourth

For the 5-limit version of this temperament, see High badness temperaments #Sfourth.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 64827/64000

Mapping: [⟨1 2 3 3], ⟨0 -19 -31 -9]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/48 = 26.287 ¢

Optimal ET sequence: 45, 46, 91, 137d

Badness (Sintel): 3.120

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 4375/4374

Mapping: [⟨1 2 3 3 4], ⟨0 -19 -31 -9 -25]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/48 = 26.286 ¢

Optimal ET sequence: 45e, 46, 91e, 137de

Badness (Sintel): 1.788

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 325/324, 441/440

Mapping: [⟨1 2 3 3 4 4], ⟨0 -19 -31 -9 -25 -14]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/48 = 26.310 ¢

Optimal ET sequence: 45ef, 46, 91ef, 137def

Badness (Sintel): 1.366

Sfour

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2401/2376, 4375/4374

Mapping: [⟨1 2 3 3 3], ⟨0 -19 -31 -9 21]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/48 = 26.246 ¢

Optimal ET sequence: 45, 46, 91, 137d

Badness (Sintel): 2.531

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 364/363, 385/384, 4375/4374

Mapping: [⟨1 2 3 3 3 3], ⟨0 -19 -31 -9 21 32]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/48 = 26.239 ¢

Optimal ET sequence: 45, 46, 91, 137d

Badness (Sintel): 2.144

Trideci

For the 5-limit version of this temperament, see 13th-octave 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 tridecatonic temperament, but with the ragisma (4375/4374) rather than the octagar (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: [⟨13 0 -11 57], ⟨0 1 2 -1]]

Optimal tuning (POTE): ~256/245 = 92.3077 ¢, ~3/2 = 699.1410 ¢

Optimal ET sequence: 26, 65, 91, 156d, 247cdd

Badness (Sintel): 4.671

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 385/384, 4375/4374

Mapping: [⟨13 0 -11 57 45], ⟨0 1 2 -1 0]]

Optimal tuning (POTE): ~22/21 = 92.3077 ¢, ~3/2 = 699.6179 ¢

Optimal ET sequence: 26, 65, 91, 156d, 247cdde

Badness (Sintel): 2.796

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 245/242, 325/324, 385/384

Mapping: [⟨13 0 -11 57 45 48], ⟨0 1 2 -1 0 0]]

Optimal tuning (POTE): ~22/21 = 92.3077 ¢, ~3/2 = 699.2969 ¢

Optimal ET sequence: 26, 65f, 91f, 156dff

Badness (Sintel): 2.164

Counterorson

Counterorson tempers out the [147 -103 7⟩ comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the semicomma family, 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on 53edo.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [154 -54 -21 -7⟩

Mapping: [⟨1 0 -21 85], ⟨0 7 103 -363]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~[66 -23 -9 -3⟩ = 271.7113 ¢

Optimal ET sequence: 53, …, 1612, 1665, 1718

Badness (Sintel): 7.916

References

  1. ↑ [1]: "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."
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