Octagar family

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.

The octagar family of rank-3 temperaments tempers out the octagar comma, 4000/3969.

For rank-2 temperaments that temper out the octagar comma, see Octagar temperaments.

Octagar

Octagar is generated by a perfect fifth and a minor sixth of ~63/40, two of which make ~5/2, and three make an interval short of two octaves by a septimal comma.

While many 7-limit rank-3 temperaments exhibit a tendency towards tuning flat, a slight sharp tendency is often preferred instead. Octagar provides this; for instance the 7-odd-limit minimax tuning has 3's and 7's 2.245 cents sharp, with just 5's.

This temperament is catalogued as octagari in Graham Breed's Temperament Finder.

Subgroup: 2.3.5.7

Comma list: 4000/3969

Mapping: [⟨1 0 1 4], ⟨0 1 0 -2], ⟨0 0 2 3]]

Mapping generators: ~2, ~3, ~63/40

Mapping to lattice: [⟨0 -1 -2 -1], ⟨0 -1 0 2]]

Lattice basis:

63/50 length = 0.8966, 21/20 length = 1.0605

Angle (63/50, 21/20) = 97.743 degrees

Optimal tunings:

• WE: ~2 = 1199.5353 ¢, ~3/2 = 703.3499 ¢, ~63/40 = 792.6380 ¢

error map: ⟨-0.465 +0.930 -1.502 +1.459]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.3764 ¢, ~63/40 = 792.6299 ¢

error map: ⟨0.000 +1.421 -1.054 +2.311]

Minimax tuning:

• 7-odd-limit

[[1 0 0 0⟩, [5/6 1/3 1/2 -1/3⟩, [0 0 1 0⟩, [5/6 -2/3 1/2 2/3⟩]

unchanged-interval (eigenmonzo) basis: 2.5.7/3

• 9-odd-limit

[[1 0 0 0⟩, [5/8 1/2 3/8 -1/4⟩, [0 0 1 0⟩, [5/4 -1 3/4 1/2⟩]

unchanged-interval (eigenmonzo) basis: 2.5.9/7

Optimal ET sequence: 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d

Badness (Sintel): 0.951

Projection pairs: 5 3969/800 7 27783/4000 to 2.3.7/5

Hobbit bases

2.3.7/5 subgroup

• 12: 50/49, 256000/250047
• 15: 256000/250047, 1029/1000
• 23: 12800000/12252303, 107163/102400

Nakika

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242

Mapping: [⟨1 0 1 4 4], ⟨0 1 0 -2 -2], ⟨0 0 2 3 4]]

Mapping to lattice: [⟨0 1 2 1 2], ⟨0 -1 0 2 2]]

Lattice basis:

11/7 length = 0.798, 22/21 length = 0.906

Angle (11/7, 22/21) = 97.747 degrees

Optimal tunings:

• WE: ~2 = 1199.6137 ¢, ~3/2 = 703.6571 ¢, ~11/7 = 791.5207 ¢

error map: ⟨-0.386 +1.316 -3.659 -2.351 +6.678]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.6783 ¢, ~11/7 = 791.5168 ¢

error map: ⟨0.000 +1.723 -3.280 -1.632 +7.393]

Optimal ET sequence: 12, 15, 26, 27e, 41, 109e

Badness (Sintel): 0.647

Projection pairs: 5 242/49 7 21296/3087 11 234256/21609 to 2.3.11/7

Associated temperament: octacot

Scales: nakika12

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 245/242

Mapping: [⟨1 0 1 4 4 2], ⟨0 1 0 -2 -2 -1], ⟨0 0 2 3 4 5]]

Optimal tunings:

• WE: ~2 = 1200.5397 ¢, ~3/2 = 702.2038 ¢, ~11/7 = 789.6574 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0977 ¢, ~11/7 = 789.5686 ¢

Optimal ET sequence: 12f, 14cf, 15, 26, 29, 41

Badness (Sintel): 0.832

Octasand

Subgroup: 2.3.5.7.11

Comma list: 540/539, 2200/2187

Mapping: [⟨1 0 1 4 -5], ⟨0 1 0 -2 7], ⟨0 0 2 3 -4]]

Optimal tunings:

• WE: ~2 = 1199.5255 ¢, ~3/2 = 703.2719 ¢, ~63/40 = 792.6321 ¢

error map: ⟨-0.475 +0.842 -1.524 +1.578 +0.108]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.2662 ¢, ~63/40 = 792.6212 ¢

error map: ⟨0.000 +1.311 -1.071 +2.505 +1.061]

Optimal ET sequence: 27e, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d

Badness (Sintel): 1.20

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 1 4 -5 0], ⟨0 1 0 -2 7 4], ⟨0 0 2 3 -4 -4]]

Optimal tunings:

• WE: ~2 = 1199.5112 ¢, ~3/2 = 703.2823 ¢, ~63/40 = 792.6491 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.3684 ¢, ~63/40 = 792.7875 ¢

Optimal ET sequence: 27e, 41, 53, 80, 94, 121, 174d, 215, 295d, 336def

Badness (Sintel): 1.03