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 tenth	5L+4s
14	8^	augmented ninth	7L+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

# generators up	Notation (1/1 = G ut, ~8/3 = C sol fa ut)	name	In L's and s's	# generators up	Notation of twenty-first inverse	name	In L's and s's
The 20-note MOS has the following intervals (from some root):
0	G ut, C sol fa ut	perfect unison, perfect eleventh	0, 7L+3s	0	G ut, C sol fa ut	perfect eleventh, “perfect” minor twenty-first	7L+3s, 14L+6s
1	G sol fa re ut, C fa	perfect octave, perfect eighteenth	5L+2s, 12L+5s	-1	C fa ut, F fa ut	perfect fourth, minor fourteenth	2L+1s, 9L+4s
2	D sol re ut, G sol fa ut	just fifth, perfect fifteenth	3L+1s, 10L+4s	-2	F fa ut, B fa	minor seventh, minor seventeenth	4L+2s, 11L+5s
3	A re, D sol la re ut	major second, perfect twelfth	1L, 8L+3s	-3	B fab, E lab	minor tenth, minor twentieth	6L+3s, 13L+6s
4	A la sol re mi, D sol	major ninth, perfect nineteenth	6L+2s, 13L+5s	-4	B mib, E la mi reb	minor third, minor thirteenth	1L+1s, 8L+4s
5	E la mi re, A la sol re	major sixth, major sixteenth	4L+1s, 11L+4s	-5	E la mi reb, A la mi reb	minor sixth, minor sixteenth	3L+2s, 10L+5s
6	B mi, E la mi re si	major third, major thirteenth	2L, 9L+3s	-6	A la sol reb, D solb	minor ninth, diminished nineteenth	5L+3s, 12L+6s
7	B si la mi, E la	major tenth, major twentieth	7L+2s, 14L+5s	-7	A reb, D sol la reb	minor second, diminished twelfth	1s, 7L+4s
8	F si mi, B si la mi	major seventh, major seventeenth	5L+1s, 12L+4s	-8	D sol re utb, G sol fa utb	diminished fifth, diminished fifteenth	2L+2s, 9L+5s
9	C si, F si mi	augmented fourth, major fourteenth	3L, 10L+3s	-9	G sol fa utb, C fab	diminished octave, diminished eighteenth	4L+3s, 11L+6s
10	G ut#, C si	augmented unison, augmented eleventh	1L-1s, 8L+2s	-10	G utb, C fa utb	diminished eleventh, diminished twenty-first	6L+4s, 13L+7s
The chromatic 17-note MOS (either 14L 20s, 20L 14s, or 34edXXI) also has the following intervals (from some root):
11	G sol re ut#, C si	augmented octave, augmented eighteenth	6L+1s, 13L+4s	-11	C fa utb, F fa utb	diminished fourth, diminished fourteenth	1L+2s, 8L+5s
12	D sol re ut#, G sol fa ut#	augmented fifth, augmented fifteenth	4L, 11L+3s	-12	F fa utb, B fab	diminished seventh, diminished seventeeth	3L+3s, 10L+6s
13	A re#, D sol re ut#	augmented second, augmented twelfth	2L-1s, 9L+2s	-13	B fab, E labb	diminished tenth, diminished twentieth	5L+4s, 12L+7s
14	A la sol re mi#, D sol#	augmented ninth, augmented nineteenth	7L+1s, 14L+4s	-14	B mibb, E la mi re sibb	diminished third, diminished thirteenth	2s, 7L+5s
15	E la mi re#, A la sol re#	augmented sixth, augmented sixteenth	5L, 12L 3s	-15	E la mi rebb, A la sol rebb	diminished sixth, diminished sixteenth	2L+3s, 9L+6s
16	B mi#, E la mi re si#	augmented third, augmented thirteenth	3L-1s, 10L+2s	-16	A sol la mi rebb, D la solbb	diminished ninth, doubly diminished nineteenth	4L+4s, 11L+7s

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	509.091	5	4	1.250
59\84	508.475	9	7	1.286
26\37	507.692	4	3	1.333
45\64	506.667	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	501.818	9	5	1.800
12\17	500.000	2	1	2.000	Basic Bolivar (Generators smaller than this are proper)
101\143	499.010	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	490.909	4	1	4.000
49\69	489.796	9	2	4.500
27\38	488.889	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

Comma list: 81/80

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

Comma list: 64/63

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.

User:Moremajorthanmajor/7L 3s (perfect eleventh-equivalent)

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