Octagar temperaments

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 octagar comma (monzo: [5 -4 3 -2⟩, ratio: 4000/3969).

Temperaments considered below are slendi and slithy. Discussed elsewhere are:

• Sida (+25/24) → Dicot family
• Rip (+36/35) → Ripple family
• Nautilus (+49/48) → Porcupine family
• Injera (+50/49 or 81/80) → Meantone family
• Augene (+64/63 or 126/125) → Augmented family
• Garibaldi (+225/224) → Schismatic family
• Octacot (+245/243) → Tetracot family
• Roman (+525/512) → Wesley family
• Superkleismic (+875/864 or 1029/1024) → Gamelismic clan
• Quartonic (+1728/1715) → Quartonic family
• Bidia (+2048/2025 or 3136/3125) → Diaschismic family
• Hamity (+2430/2401) → Amity family
• Hemikleismic (+6144/6125) → Kleismic family
• Pluto (+10976/10935) → Mirkwai clan
• Quintupole (+458752/455625) → Quintaleap family

Slendi

Slendi may be described as the 41 & 42 temperament. It has a generator of 49/48, seventeen of which give 4/3. Its ploidacot is omega-17-cot.

The name was likely derived from slendro diesis, one of the names for the interval 49/48.

Subgroup: 2.3.5.7

Comma list: 4000/3969, 84035/82944

Mapping: [⟨1 2 3 3], ⟨0 -17 -28 -8]]

mapping generators: ~2, ~49/48

Optimal tunings:

• WE: ~2 = 1200.2515 ¢, ~49/48 = 29.1870 ¢

error map: ⟨+0.252 +2.368 -2.796 -1.568]

• CWE: ~2 = 1200.0000 ¢, ~49/48 = 29.1715 ¢

error map: ⟨0.000 +2.129 -3.116 -2.198]

Optimal ET sequence: 1c, 40c, 41, 206

Badness (Sintel): 3.62

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 46656/46585

Mapping: [⟨1 2 3 3 4], ⟨0 -17 -28 -8 -22]]

Optimal tunings:

• WE: ~2 = 1199.8027 ¢, ~49/48 = 29.1751 ¢
• CWE: ~2 = 1200.0000 ¢, ~49/48 = 29.1875 ¢

Optimal ET sequence: 1ce, 40c, 41

Badness (Sintel): 2.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 245/242, 343/338

Mapping: [⟨1 2 3 3 4 4], ⟨0 -17 -28 -8 -22 -12]]

Optimal tunings:

• WE: ~2 = 1199.0915 ¢, ~49/48 = 29.1033 ¢
• CWE: ~2 = 1200.0000 ¢, ~49/48 = 29.1576 ¢

Optimal ET sequence: 1ce, 41

Badness (Sintel): 1.80

Slithy

Subgroup: 2.3.5.7

Comma list: 4000/3969, 420175/419904

Mapping: [⟨1 3 5 4], ⟨0 -19 -36 -16]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~21/20 = 89.282 ¢

Optimal ET sequence: 27, 94, 121, 215, 336d

Badness (Smith): 0.107482

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 2200/2187, 4375/4356

Mapping: [⟨1 3 5 4 8], ⟨0 -19 -36 -16 -61]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~21/20 = 89.291 ¢

Optimal ET sequence: 94, 121, 215, 336de, 551bde

Badness (Smith): 0.045895

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 540/539, 1575/1573

Mapping: [⟨1 3 5 4 8 4], ⟨0 -19 -36 -16 -61 -4]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~21/20 = 89.286 ¢

Optimal ET sequence: 94, 121, 215, 336def

Badness (Smith): 0.028492

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 352/351, 375/374, 540/539, 595/594

Mapping: [⟨1 3 5 4 8 4 9], ⟨0 -19 -36 -16 -61 -4 -66]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~21/20 = 89.286 ¢

Optimal ET sequence: 94, 121, 215, 336defg

Badness (Smith): 0.019811

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 325/324, 352/351, 361/360, 375/374, 595/594

Mapping: [⟨1 3 5 4 8 4 9 7], ⟨0 -19 -36 -16 -61 -4 -66 -37]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~19/18 = 89.288 ¢

Optimal ET sequence: 94, 121, 215, 336defg

Badness (Smith): 0.016408

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 190/189, 300/299, 323/322, 325/324, 352/351, 361/360, 484/483

Mapping: [⟨1 3 5 4 8 4 9 7 11], ⟨0 -19 -36 -16 -61 -4 -66 -37 -87]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~19/18 = 89.297 ¢

Optimal ET sequence: 94, 121i, 215

Badness (Smith): 0.014946

Tridecatonic

See also: 13th-octave temperaments § Tridecatonic

The tridecatonic temperament (26 & 39) has a period of 1/13 octave. It tempers out the tridecalimma, [-11 26 -13⟩ in the 5-limit. There are some extensions for 7-limit, including tridecatonic (26 & 39d, tempering out 4000/3969), trideci (26 & 39, tempering out 4375/4374), and bosonic (26 & 130, tempering out 321489/320000).

Subgroup: 2.3.5.7

Comma list: 4000/3969, 8748/8575

Mapping: [⟨13 21 31 37], ⟨0 -1 -2 -1]]

Optimal tuning (POTE): ~21/20 = 92.3077 ¢, ~3/2 = 701.9605 ¢

Optimal ET sequence: 26, 39d, 65d

Badness (Smith): 0.139213

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 1944/1925, 2560/2541

Mapping: [⟨13 21 31 37 45], ⟨0 -1 -2 -1 0]]

Optimal tuning (POTE): ~21/20 = 92.3077 ¢, ~3/2 = 701.7028 ¢

Optimal ET sequence: 26, 39d, 65d

Badness (Smith): 0.063257

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 144/143, 275/273, 1040/1029

Mapping: [⟨13 21 31 37 45 49], ⟨0 -1 -2 -1 0 -2]]

Optimal tuning (POTE): ~21/20 = 92.3077 ¢, ~3/2 = 701.3573 ¢

Optimal ET sequence: 26, 39df, 65d

Badness (Smith): 0.037851

Dhaivatic

Subgroup: 2.3.5.7

Comma list: 4000/3969, 2100875/2097152

Mapping: [⟨1 -4 -3 6], ⟨0 21 20 -12]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 319.198 ¢

Optimal ET sequence: 15, 64c, 79, 94, 203, 297

Badness (Smith): 0.168373

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1331/1323, 4000/3969

Mapping: [⟨1 -4 -3 6 0], ⟨0 21 20 -12 13]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 319.197 ¢

Optimal ET sequence: 15, 64c, 79, 94, 203, 297c

Badness (Smith): 0.051216

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 275/273, 325/324, 385/384, 1331/1323

Mapping: [⟨1 -4 -3 6 0 -8], ⟨0 21 20 -12 13 44]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 319.165 ¢

Optimal ET sequence: 15, 49bceff, 64cf, 79, 94

Badness (Smith): 0.037468