## Sagittal notation for Sunvar24 (19/16) system

mschulter
Posts: 4
Joined: Thu Jul 27, 2017 10:05 am

### Sagittal notation for Sunvar24 (19/16) system

This is a lattice for Sunvar-19, a variation on Scott Dakota's Sun 19 Constant Structure (CS) tuning. My idea was inspired by Scott's observation that his Sun-19 theme could have variations where the goal was to set the regular minor third (-3 fifths) to a just or near-just 19/16 (297.513 cents).

Starting from this premise, I came up with the idea of two chains of fifths at 701.350 cents, spaced at 64.171 cents so that the tempered 9/8 (202.700 cents) plus this spacing yields a pure 7/6 (266.871 cents).

The spacing can represent either 28/27 (62.961 cents), as in the first lattice with D-E/|\ (9/8-7/6); or 27/26 (65.337 cents), as in the second lattice with E\|/-E (13/12-9/8). These JI thirdtone steps differ by 729/728 (2.376 cents). The spacing may also represent the complex ratio of 256/247 (61.959 cents), curiously smaller than 28/27 by 1729/1728 (1.002 cents), or almost exactly a cent, e.g. in the first lattice F-F/|\ (19/16-16/13); and in the second lattice, F#\|/-F# (39/32-24/19).

To find a counterpart to Scott's 19-note set, I have indicated on the lattice diagrams below a 10-note subset of the lower chain of fifths (Eb-F# in the first lattice, Eb\|/-F#\|/ in the second) and a 9-note subset of the upper chain (C/|\-G#/|\ and C-G# respectively) which together would yield such a 19-note subset.

This Sunvar-19 shading has close approximations for 19/16 and related ratios, while keeping ratios of 2-3-7 (e.g. 7/6, 9/7, 7/4) near-just, as well as some basic ratios of 13 (e.g. 13/9-13/12-13/8-39/32-117/64).

I should add that the original JI form of Scott's Sun-19 may be derived from two chains of pure 3:2 fifths (701.955 cents) at a pure 7:6 apart. He encourages all kinds of variations, just and tempered.
Lattice for Sunvar-24, 1/1 = D (on lower 12-note chain of fifths)

Subset used in Sunvar-19 after Scott Dakota's Sun-19
|----------------------------------------------------|
+1.9    +1.3    +0.6   -0.04   -0.6   -1.2  +0.6  just  -0.6  -1.2   -1.8   -2.4
157.4   858.8   360.1  1061.5  562.8  64.2 765.5 266.9  968.2 469.6 1170.9  672.3
Eb/|\   Bb/|\   F/|\    C/|\   G/|\   D/|\  A/|\  E/|\  B/|\  F#/|\  C#/|\  G#/|\
128/117--64/39---16/13--24/13--18/13--27/26-14/9---7/6---7/4---21/16--63/32-189/128
/      /      /      /     /     /     /      /      /      /
/      /      /      /     /     /     /      /      /      /
/      /      /      /     /     /     /      /      /      /
19/18--19/12--19/16--16/9---4/3---1/1---3/2----9/8---32/19--24/19--36/19--27/19
Eb     Bb      F     C      G     D     A      E      B     F#     C#     G#
93.3   794.6  296.0 997.3  498.7   0   701.3 202.7   904.0 405.4  1106.7  608.1
-0.4   +1.0   -1.6  +1.2   +0.6        -0.6   +1.2   +1.6  -1.0    +0.4   -0.3
|------------------------------------------------------------|
Subset used in Sunvar-19 after Scott Dakota's Sun-19

Horizontal lines = tempered 3:2 (703.711 cents)
Vertical lines = tempered 7:6 (264.844 cents)
Quadrangles from lower left = tempered 12:14:18:21
Generators of Sunvar-24 temperament: (2/1, 701.350, 64.171)

! sunvar24-19_16.scl
!
Variation on Scott Dakota's Sun 19 (24): optimized for 16:19:24 (2/1, 701.350, 64.171)
24
!
64.17091
93.25000
157.42091
202.70000
266.87091
295.95000
360.12091
405.40000
469.57091
498.65000
562.82091
608.10000
672.27091
701.35000
765.52091
794.60000
858.77091
904.05000
968.22091
997.30000
1061.47091
1106.75000
1170.92091
2/1

Lattice for Sunvar-24, 1/1 = D (on upper 12-note chain of fifths)

Subset used in Sunvar-19 after Scott Dakota's Sun-19
|------------------------------------------------------|
-0.4   +1.0   -1.6  +1.2   +0.6         -0.6   +1.2   +1.6  -1.0    +0.4   -0.3
93.3   794.6  296.0 997.3  498.7    0   701.3  202.7  904.0 405.4  1106.7  608.1
Eb     Bb      F     C      G      D     A      E      B     F#     C#     G#
19/18--19/12--19/16--16/9----4/3----1/1---3/2---9/8---32/19--24/19--36/19--27/19
/      /      /      /     /     /      /      /      /      /
/      /      /      /     /     /      /      /      /      /
/      /      /      /     /     /      /      /      /      /
64/63--32/21---8/7---12/7---9/7--52/27--13/9---13/12--13/8---39/32--117/64--351/256
Eb\|/--Bb\|/--F\|/---C\|/--G\|/--D\|/---A\|/----E\|/--B\|/---F#\|/---C#\|/---G#\|/
29.1   730.4  231.8 933.1 434.5 1135.8  637.2  138.5  839.9  341.2  1042.6   543.9
+1.8   +1.2   +0.6   just -0.6   +1.2   +0.6   -0.04  -0.6   -1.3    -1.9     -2.5
|------------------------------------------------------------|
Subset used in Sunvar-19 after Scott Dakota's Sun-19

! sunvar24_dup.scl
!
Sunvar24, 1/1=D on upper chain of fifths
24
!
29.07909
93.25000
138.52909
202.70000
231.77909
295.95000
341.22909
405.40000
434.47909
498.65000
543.92909
608.10000
637.17909
701.35000
730.42909
794.60000
839.87909
904.05000
933.12909
997.30000
1042.57909
1106.75000
1135.82909
2/1