Diaschismic rank-3 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 diaschismic rank-3 family of rank-3 temperaments tempers out the diaschisma, 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.
Rank-3 diaschismic
Subgroup: 2.3.5.7
Mapping: [⟨2 0 11 0], ⟨0 1 -2 0], ⟨0 0 0 1]]
- mapping generators: ~45/32, ~3, ~7
- WE: ~45/32 = 599.4107 ¢, ~3/2 = 704.2059 ¢, ~7/4 = 971.1782 ¢
- error map: ⟨-1.179 +1.072 +1.150 -0.005]
- CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.9585 ¢, ~7/4 = 971.2952 ¢
- error map: ⟨0.000 +3.003 +3.769 +2.469]
Optimal ET sequence: 10, 12, 22, 34d, 46, 58, 68, 80, 126, 206cd, 332bccd
Badness (Sintel): 1.95
Varda
Varda tempers out 176/175, 896/891 as well as 9801/9800, and is the main extension of rank-3 diaschismic to the 11-limit. Notice the identity 2048/2025 = (176/175)⋅(896/891). In the natural 13-limit extension, 351/350, 352/351 and 364/363 are further tempered out, and in the 17-limit, 136/135, 256/255 and 289/288.
Subgroup: 2.3.5.7.11
Comma list: 176/175, 896/891
Mapping: [⟨2 0 11 0 14], ⟨0 1 -2 0 -4], ⟨0 0 0 1 1]]
Mapping to lattice: [⟨0 1 -2 2 -2], ⟨0 0 0 -1 -1]]
Lattice basis:
- 3/2, 9/7
- Angle (3/2, 9/7) = 82.289 degrees
- WE: ~45/32 = 599.4084 ¢, ~3/2 = 703.9071 ¢, ~7/4 = 971.8498 ¢
- error map: ⟨-1.183 +0.769 +1.731 +0.657 -1.013]
- CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.6576 ¢, ~7/4 = 971.9790 ¢
- error map: ⟨0.000 +2.703 +4.371 +3.153 +2.031]
- [[1 0 0 0 0⟩, [11/8 1/2 -1/4 0 0⟩, [11/4 -1 1/2 0 0⟩, [-1/8 3/2 -5/4 0 1⟩, [11/8 -1/2 -1/4 0 1⟩]
- unchanged-interval (eigenmonzo) basis: 2.9/5.11/3
Optimal ET sequence: 12, 22, 34d, 46, 58, 68, 80, 126, 184c, 206cd, 264bccde
Badness (Sintel): 0.875
- Scales
- Varda[22] hobbit transversal:
- 28/27, 16/15, 11/10, 9/8, 32/27, 6/5, 5/4, 9/7, 4/3, 15/11, 45/32, 22/15, 3/2, 14/9, 8/5, 5/3, 27/16, 16/9, 20/11, 15/8, 27/14, 2
- Varda[24] hobbit transversal
- 28/27, 16/15, 11/10, 9/8, 7/6, 6/5, 40/33, 5/4, 9/7, 4/3, 15/11, 45/32, 16/11, 3/2, 14/9, 8/5, 33/20, 27/16, 7/4, 16/9, 20/11, 15/8, 27/14, 2
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350, 364/363
Mapping: [⟨2 0 11 0 14 24], ⟨0 1 -2 0 -4 -7], ⟨0 0 0 1 1 1]]
Optimal tunings:
- WE: ~45/32 = 599.4000 ¢, ~3/2 = 703.5375 ¢, ~7/4 = 971.9244 ¢
- CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.2888 ¢, ~7/4 = 972.0576 ¢
Optimal ET sequence: 12f, 22, 34d, 46, 58, 80, 104c, 138cde, 242ccde
Badness (Sintel): 0.908
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 136/135, 176/175, 256/255, 351/350
Mapping: [⟨2 0 11 0 14 24 5], ⟨0 1 -2 0 -4 -7 1], ⟨0 0 0 1 1 1 0]]
Optimal tunings:
- WE: ~17/12 = 599.5734 ¢, ~3/2 = 703.8639 ¢, ~7/4 = 971.4469 ¢
- CWE: ~17/12 = 600.0000 ¢, ~3/2 = 704.3790 ¢, ~7/4 = 971.6313 ¢
Optimal ET sequence: 12f, 22, 34d, 46, 58, 80, 104c, 126, 138cde
Badness (Sintel): 0.841
Julius
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 325/324, 896/891
Mapping: [⟨2 0 11 0 14 -18], ⟨0 1 -2 0 -4 8], ⟨0 0 0 1 1 0]]
Optimal tunings:
- WE: ~45/32 = 599.5350 ¢, ~3/2 = 704.6999 ¢, ~7/4 = 972.2975 ¢
- CWE: ~45/32 = 600.0000 ¢, ~3/2 = 705.1906 ¢, ~7/4 = 972.3332 ¢
Optimal ET sequence: 12, 22f, 34d, 46, 68, 80, 148d, 228bcd
Badness (Sintel): 1.66
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 136/135, 176/175, 256/255, 325/324
Mapping: [⟨2 0 11 0 14 -18 5], ⟨0 1 -2 0 -4 8 1], ⟨0 0 0 1 1 0 0]]
Optimal tunings:
- WE: ~17/12 = 599.6247 ¢, ~3/2 = 704.7816 ¢, ~7/4 = 971.8531 ¢
- CWE: ~17/12 = 600.0000 ¢, ~3/2 = 705.1801 ¢, ~7/4 = 971.9669 ¢
Optimal ET sequence: 12, 22f, 34d, 46, 68, 80, 126, 148d
Badness (Sintel): 1.18
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 136/135, 176/175, 190/189, 256/255, 286/285
Mapping: [⟨2 0 11 0 14 -18 5 -13], ⟨0 1 -2 0 -4 8 1 5], ⟨0 0 0 1 1 0 0 1]]
Optimal tunings:
- WE: ~17/12 = 599.6179 ¢, ~3/2 = 704.8484 ¢, ~7/4 = 972.3437 ¢
- CWE: ~17/12 = 600.0000 ¢, ~3/2 = 705.2584 ¢, ~7/4 = 972.4875 ¢
Optimal ET sequence: 12, 22fh, 34dh, 46, 68, 80, 148d, 228bcd
Badness (Sintel): 1.22
Augustus
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1232/1215
Mapping: [⟨2 0 11 0 3], ⟨0 1 -2 0 3], ⟨0 0 0 1 -1]]
- WE: ~45/32 = 599.7497 ¢, ~3/2 = 705.5140 ¢, ~7/4 = 967.5010 ¢
- error map: ⟨-0.501 +3.058 +0.906 -2.326 -3.528]
- CWE: ~45/32 = 600.0000 ¢, ~3/2 = 705.7687 ¢, ~7/4 = 967.8291 ¢
- error map: ⟨0.000 +3.814 +2.149 -0.997 -1.841]
- [[1 0 0 0 0⟩, [11/8 1/2 -1/4 0 0⟩, [11/4 -1 1/2 0 0⟩, [45/16 3/4 -3/8 1/2 -1/2⟩, [45/16 3/4 -3/8 -1/2 1/2⟩]
- unchanged-interval (eigenmonzo) basis: 2.9/5.11/7
Optimal ET sequence: 10, 12e, 22, 46, 68, 114, 170b
Badness (Sintel): 1.90
Tiberius
Subgroup: 2.3.5.7.11
Comma list: 441/440, 2048/2025
Mapping: [⟨2 0 11 0 -17], ⟨0 1 -2 0 4], ⟨0 0 0 1 2]]
- WE: ~45/32 = 599.4432 ¢, ~3/2 = 703.7469 ¢, ~7/4 = 968.7525 ¢
- error map: ⟨-0.501 +3.058 +0.906 -2.326 -3.528]
- CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.4801 ¢, ~7/4 = 968.9703 ¢
- error map: ⟨0.000 +3.814 +2.149 -0.997 -1.841]
- [[1 0 0 0 0⟩, [11/8 1/2 -1/4 0 0⟩, [11/4 -1 1/2 0 0⟩, [0 0 0 1 0⟩, [-3 2 -1 2 0⟩]
- unchanged-interval (eigenmonzo) basis: 2.9/5.7
Optimal ET sequence: 10, 12, 22e, 34e, 36e, 44d, 46, 58, 104c, 160, 162ce, 172ce, 218ce
Badness (Sintel): 2.05
Claudius
Subgroup: 2.3.5.7.11
Comma list: 540/539, 2048/2025
Mapping: [⟨2 0 11 0 15], ⟨0 1 -2 0 1], ⟨0 0 0 1 -2]]
- WE: ~45/32 = 599.2988 ¢, ~3/2 = 703.9668 ¢, ~7/4 = 972.7468 ¢
- error map: ⟨-1.402 +0.609 +0.845 +1.116 +0.845]
- CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.7949 ¢, ~7/4 = 974.0012 ¢
- error map: ⟨0.000 +2.840 +4.097 +5.175 +5.475]
- [[1 0 0 0 0⟩, [9/7 0 -3/7 2/7 1/7⟩, [41/14 0 6/7 -4/7 -2/7⟩, [41/14 0 -1/7 3/7 -2/7⟩, [41/14 0 -1/7 -4/7 5/7⟩]
- unchanged-interval (eigenmonzo) basis: 2.7/5.11/5
Optimal ET sequence: 10, 12e, 22, 46e, 48c, 58, 80, 138cde, 218cde, 286bcddeee, 366bccdddeee
Badness (Sintel): 2.37