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

Posted:

**Mon Jul 31, 2017 10:33 am**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.

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 Two adjacent quadrangles = tempered 1-3-7-9 hexany 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