User:Moremajorthanmajor/7L 3s (perfect eleventh-equivalent)
7L 3s<perfect eleventh> (sometimes called Bolivar or Choralic) refers to a non-octave MOS scale family with a period of a perfect eleventh and which has 7 large and 3 small steps. These scales are the sister of diaquadic with the melodic spacing of diatonic scales. A pathological trait these scales exhibit is that normalization to edo collapses the range for the bright generator to the octave.
Modes
The modes contain fundamental chords with notes such that they convert a tritone substitution into a diatonic chord substitution.
- LLLsLLsLLs 9|0 (Lydian ♮11)
- LLsLLLsLLs 8|1 (Major, Ionian)
- LLsLLsLLLs 7|2 (Mixolydian)
- LLsLLsLLsL 6|3 (Mahur)
- LsLLLsLLsL 5|4 (Dorian)
- LsLLsLLLsL 4|5 (Minor, Aeolian)
- LsLLsLLsLL 3|6 (Aeolian b9)
- sLLLsLLsLL 2|7 (Phrygian)
- sLLsLLLsLL 1|8 (Locrian)
- sLLsLLsLLL 0|9 (Locrian b8)
Intervals
The generator (g) will fall between 480 cents (2\5 - two degrees of 5edo) and 514 cents (2\5 - two degrees of 5edo), hence a perfect fourth.
2g, then, will fall between 960 cents (4\5) and 1029 cents (6\7), the range of minor sevenths.
The "large step" will fall between 171 cents (1\7) and 240 cents (1\5), the range of major seconds.
The "small step" will fall between 0 cents and 171 cents, sometimes sounding like a submajor second, and sometimes sounding like a quartertone or smaller microtone.
| # generators up | Notation (1/1 = 0) | name | In L's and s's | # generators up | Notation of eleventh inverse | name | In L's and s's |
|---|---|---|---|---|---|---|---|
| The 10-note MOS has the following intervals (from some root): | |||||||
| 0 | 0 | perfect unison | 0 | 0 | 0 | perfect eleventh | 7L+3s |
| 1 | 7 | perfect octave | 5L+2s | -1 | 3 | perfect fourth | 2L+1s |
| 2 | 4 | just fifth | 3L+1s | -2 | 6 | minor seventh | 4L+2s |
| 3 | 1 | major second | 1L | -3 | 9v | minor tenth | 6L+3s |
| 4 | 8 | major ninth | 6L+2s | -4 | 2v | minor third | 1L+1s |
| 5 | 5 | major sixth | 4L+1s | -5 | 5v | minor sixth | 3L+2s |
| 6 | 2 | major third | 2L | -6 | 8v | minor ninth | 5L+3s |
| 7 | 9 | major tenth | 7L+2s | -7 | 1v | minor second | 1s |
| 8 | 6^ | major seventh | 5L+1s | -8 | 4v | diminished fifth | 2L+2s |
| 9 | 3^ | augmented fourth | 3L | -9 | 7v | diminished octave | 4L+3s |
| 10 | 0^ | augmented unison | 1L-1s | -10 | 0v | diminished eleventh | 6L+4s |
| The chromatic 17-note MOS (either 7L 10s, 10L 7s, or 17edXI) also has the following intervals (from some root): | |||||||
| 11 | 7^ | augmented octave | 6L+1s | -11 | 3v | diminished fourth | 1L+2s |
| 12 | 4^ | augmented fifth | 4L | -12 | 6v | diminished seventh | 3L+3s |
| 13 | 1^ | augmented second | 2L-1s | -13 | 9w | diminished ninth | 5L+4s |
| 14 | 8^ | augmented ninth | 8L+1s | -14 | 2w | diminished third | 2s |
| 15 | 5^ | augmented sixth | 5L | -15 | 5w | diminished sixth | 2L+3s |
| 16 | 2^ | augmented third | 3L-1s | -16 | 8w | diminished ninth | 4L+4s |
Scale tree
The generator range reflects two extremes: one where L = s (3\10), and another where s = 0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible ~ed8/3s, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between ~ed8/3 would be (3+2)\(10+7) = 5\17 – five degrees of 17edXI:
| Generator | Cents | L | s | L/s | Comments |
|---|---|---|---|---|---|
| 7\10 | 514.286 | 1 | 1 | 1.000 | |
| 40\57 | 510.000 | 6 | 5 | 1.200 | |
| 33\47 | 509.09 | 5 | 4 | 1.250 | |
| 59\84 | 508.475 | 9 | 7 | 1.286 | |
| 26\37 | 507.692 | 4 | 3 | 1.333 | |
| 45\64 | 506.6 | 7 | 5 | 1.400 | |
| 19\27 | 505.263 | 3 | 2 | 1.500 | L/s = 3/2 |
| 50\71 | 504.000 | 8 | 5 | 1.600 | |
| 31\44 | 503.226 | 5 | 3 | 1.667 | |
| 43\61 | 502.326 | 7 | 4 | 1.750 | |
| 55\78 | 501.81 | 9 | 5 | 1.800 | |
| 12\17 | 500.000 | 2 | 1 | 2.000 | Basic Bolivar
(Generators smaller than this are proper) |
| 101\143 | 499.0099 | 17 | 8 | 2.125 | |
| 89\126 | 498.876 | 15 | 7 | 2.143 | |
| 77\109 | 498.701 | 13 | 6 | 2.167 | |
| 65\92 | 498.452 | 11 | 5 | 2.200 | |
| 53\75 | 498.113 | 9 | 4 | 2.250 | |
| 41\58 | 497.561 | 7 | 3 | 2.333 | |
| 29\41 | 496.552 | 5 | 2 | 2.500 | |
| 46\65 | 495.652 | 8 | 3 | 2.667 | |
| 17\24 | 494.118 | 3 | 1 | 3.000 | L/s = 3/1 |
| 73\103 | 493.151 | 13 | 4 | 3.250 | |
| 56\79 | 492.857 | 10 | 3 | 3.333 | |
| 39\55 | 492.308 | 7 | 2 | 3.500 | |
| 61\86 | 491.803 | 11 | 3 | 3.667 | |
| 22\31 | 490.90 | 4 | 1 | 4.000 | |
| 49\69 | 489.796 | 9 | 2 | 4.500 | |
| 27\38 | 488.8 | 5 | 1 | 5.000 | |
| 32\45 | 487.500 | 6 | 1 | 6.000 | |
| 5\7 | 480.000 | 1 | 0 | → inf |
The scale produced by stacks of 5\17 is the 12edo diatonic scale.
Other compatible ~ed8/3s include: ~37ed8/3, ~27ed8/3, ~44ed8/3, ~41ed8/3, ~24ed8/3, ~31ed8/3.
You can also build this scale by equally dividing frequency ratio 8:3 which is not a member of an edo or stacking frequency ratio 4:3 which is not a member of an equal division of it within it.
Rank-2 temperaments
The Bolivar rank-2 temperament spells its major tetrad 4:5:6:8 or 14:18:21:28root-3(2g-p)-(2g-p)-(1g) (p = 8/3, g = 2/1) and its minor tetrad 6:7:9:12 or 10:12:15:20 root-2(p-2g)-(2g-p)-(1g) (p = 8/3, g = 2/1). Basic 17ed8/3 fits both interpretations.
Bolivar-Meantone
Subgroup: 8/3.2.5/4
POL2 generator: ~2/1 = 1196.3254
Mapping: [⟨1 0 -3], ⟨0 1 6]]
Optimal ET sequence: 17ed8/3, 27ed8/3, 44ed8/3
Bolivar-Superpyth
Subgroup: 8/3.2.7/6
POL2 generator: ~2/1 = 1206.6167
Mapping: [⟨1 0 2], ⟨0 1 -4]]
Optimal ET sequence: 17ed8/3, 24ed8/3, 31ed8/3, 38ed8/3
7-note subsets
If you stop the chain at 7 tones, you have a heptatonic scale of the form 3L 4s:
L s s L s L s
The large steps here consist of L+s of the 10-tone system, and the small step is the same as L.
Tetrachordal structure
Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a tetrachordal scale.
See also
7L 3s (8/3-equivalent) - idealized tuning
14L 6s (7/1-equivalent) - Guidotonic dominant Archytas tuning
14L 6s (78/11-equivalent) - Guidotonic dominant Neogothic tuning
14L 6s (36/5-equivalent) - Guidotonic dominant Meantone tuning