Kite's thoughts on the V-I cadence in higher prime limits

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Consider the V7-I cadence in 5-limit JI. We can use parsimonious voice-leading to create small melodic steps. This table uses octave numbers:

V7-I Cadence in 5-limit JI
V7 chord I chord melodic steps
F3 8/3 yE3 5/2 (g2) 15/16 -112¢
D3 9/4 y2 10/9 182¢
C3 2/1 (w2) 8/9 -204¢
yB2 15/8 g2 16/15 112¢
G2 3/2 G2 3/2 w1 1/1 0¢
C2 1/1

Both F and D resolve to yE. D resolves both up to yE and down to C.

Altering the V chord to the 7 limit creates a cadence in which one harmonic series segment resolves to another harmonic series segment a fifth lower. All melodic steps are delta-1 ratios.

Harmonic V7-I Cadence
V(4:5:6:7) to I(2:3:4:5)
V7 chord I chord melodic steps
7 zF3 21/8 5 yE3 5/2 (zg2) 20/21 -84¢
6 D3 9/4 y2 10/9 182¢
4 C3 2/1 (w2) 8/9 -204¢
5 yB2 15/8 g2 16/15 112¢
4 G2 3/2 3 G2 3/2 w1 1/1 0¢
2 C2 1/1

Going further up the harmonic series results in a pentad resolving to a tetrad.

Harmonic V9-I7 Cadence
V(4::10) to I(2::8)
V9 chord I7 chord melodic steps
10 yB3 15/2 8 C4 8/1 g2 16/15 112¢
7 zBb3 7/1 (ry1) 14/15 -119¢
9 A3 27/4 z2 28/27 63¢
8 G3 6/1 6 G3 6/1
7 zF3 21/4 5 yE3 5/1 (zg2) 20/21 -84¢
6 D3 9/2 y2 10/9 182¢
4 C3 4/1 (w2) 8/9 -204c
5 yB2 15/4 g2 16/15 112¢
4 G3 3/1 3 G3 3/1
2 C3 2/1

Continuing up to the 16th harmonic:

V(1::16) to I(1::12) Cadence
Harmonic Series on 3/4 and 1/1
V chord I chord melodic steps
16 G4 12/1 12 G4 12/1
15 yF#4 45/4 11 1oF#4 11/1 (1uy1) 44/45 -39¢
14 zF4 21/2 1or1 22/21 80¢
10 yE4 10/1 (zg2) 20/21 -84¢
13 3oE4 39/4 3uy1 40/39 44¢
12 D4 9/1 9 D4 9/1
11 1oC#4 33/4 8 C4 8/1 (1o1) 32/33 -53¢
10 yB3 15/2 g2 16/15 112¢
7 zBb3 7/1 (ry1) 14/15 -119¢
9 A3 27/4 z2 28/27 63¢
8 G3 6/1 6 G3 6/1
7 zF3 21/4 5 yE3 5/1 (zg2) 20/21 -84¢
6 D3 9/2 y2 10/9 182¢
4 C3 4/1 (w2) 8/9 -204¢
5 yB2 15/4 g2 16/15 112¢
4 G3 3/1 3 G3 3/1
3 D3 9/4 2 C3 2/1 (w2) 8/9 -204¢
2 G2 3/2 w4 4/3 498¢
1 C2 1/1 (w5) 2/3 -702¢
1 G1 3/4 w4 4/3 498¢

Every 4th note of the V chord coincides with every 3rd note of the I chord, analogous to a 4-against-3 cross rhythm. The melodic steps between 2 coinciding notes make a rough sLLs pattern, where L/s is about 2:1. In practice, many of the even harmonics would be omitted, to avoid overly narrow intervals.

Continuing on up to the 32nd harmonic:

V(1::32) to I(1::24) Cadence
Harmonic Series on 3/4 and 1/1
V chord I chord melodic steps
32 G5 24/1 24 G5 24/1
31 31oF#5 93/4 23 23oGb5 23/1 (31o23u-2) 92/93 -19¢
30 yF#5 45/2 23og2 46/45 38¢
22 1oF#5 22/1 (1uy1) 44/45 -39¢
29 29oF5 87/4 29u1o1 88/87 20¢
28 zF5 21/1 21 zF5 21/1
27 E5 81/4 20 yE5 20/1 (g1) 80/81 -22¢
26 3oE5 39/2 3uy1 40/39 44¢
19 19oEb5 19/1 (19u3o1) 38/39 -45¢
25 yyD#5 75/4 19ogg2 76/75 23¢
24 D5 18/1 18 D5 18/1
23 23oDb5 69/4 17 17oDb5 17/1 (23o17u1) 68/69 -25¢
22 1oC#5 33/2 17o1u2 34/33 52¢
16 C5 16/1 (1o1) 32/33 -53¢
21 zC5 63/4 r1 64/63 27¢
20 yB4 15/1 15 yB4 15/1
19 19oBb4 57/4 14 zBb4 14/1 (19or1) 56/57 -31¢
18 A4 27/2 z2 28/27 63¢
13 3oA4 13/1 (3u1) 26/27 -65¢
17 17oAb4 51/4 17u3o1 52/51 34¢
16 G4 12/1 12 G4 12/1
15 yF#4 45/4 11 1oF#4 11/1 (1uy1) 44/45 -39¢
14 zF4 21/2 1or1 22/21 80¢
10 yE4 10/1 (zg2) 20/21 -84¢
13 3oE4 39/4 3uy1 40/39 44¢
12 D4 9/1 9 D4 9/1
11 1oC#4 33/4 8 C4 8/1 (1o1) 32/33 -53¢
10 yB3 15/2 g2 16/15 112¢
7 zBb3 7/1 (ry1) 14/15 -119¢
9 A3 27/4 z2 28/27 63¢
8 G3 6/1 6 G3 6/1
7 zF3 21/4 5 yE3 5/1 (zg2) 20/21 -84¢
6 D3 9/2 y2 10/9 182¢
4 C3 4/1 (w2) 8/9 -204¢
5 yB2 15/4 g2 16/15 112¢
4 G3 3/1 3 G3 3/1
3 D3 9/4 2 C3 2/1 (w2) 8/9 -204¢
2 G2 3/2 w4 4/3 498¢
1 C2 1/1 (w5) 2/3 -702¢
1 G1 3/4 w4 4/3 498¢

The 4:3 cross rhythm results from the root of the V chord (G1) being 4/3 below the root of the I chord (C2). If instead we use C1, 3/2 below G1, we get a 3:2 cross-rhythm:

Harmonic V-I9 Cadence
Harmonic Series on 3/2 and 1/1
V chord I9 chord melodic steps
6 D4 9/1 9 D4 9/1
5 yB3 15/2 8 C4 8/1 g2 16/15 112¢
7 zBb3 7/1 (ry1) 14/15 -119¢
4 G3 6/1 6 G3 6/1
3 D3 9/2 5 yE3 5/1 y2 10/9 182¢
4 C3 4/1 (w2) 8/9 -204¢
2 G2 3/1 3 G2 3/1
1 G1 3/2 2 C2 2/1 w4 4/3 498¢
1 C1 1/1 (w5) 2/3 -702¢

Unfortunately, the V chord is lower prime-limit than the I chord, which feels backwards. Furthermore, the V chord is a triad and the I9 chord is a pentad. We expect the more complex chord to resolve to the simpler chord. Therefore the first chord in the cadence should always have a lower root. To achieve this, we can swap the order of the two chords to make a IV-I cadence:

Harmonic IV9-I Cadence
Harmonic Series on 2/3 and 1/1
IV9 chord I chord melodic steps
9 G4 6/1 6 G4 6/1
8 F4 16/3 5 yE4 5/1 (g2) 15/16 -112¢
7 zEb4 14/3 ry1 15/14 119¢
6 C4 4/1 4 C4 4/1
5 yA3 10/3 3 G3 3/1 (y2) 9/10 -182¢
4 F3 8/3 w2 9/8 204¢
3 C3 2/1 2 C3 2/1
2 F2 4/3 1 C2 1/1 (w4) 3/4 -498¢
1 F1 2/3 w5 3/2 702¢

Going up to the 36th harmonic:

IV(1::36) to I(1::24) Cadence
Harmonic Series on 2/3 and 1/1
IV chord I chord melodic steps
36 G6 23/1 24 G6 24/1
35 zyG6 70/3 23 23oGb6 23/1 (23uzy1) 69/70 -25¢
34 17oGb6 68/3 23o17u1 69/68 25¢
33 1oF#6 22/1 22 1oF#6 22/1
32 F6 64/3 21 zF6 21/1 (r1) 63/64 -27¢
31 31oE6 62/3 31uz2 63/62 28¢
30 yE6 20/1 20 yE6 20/1
29 29oEb6 58/3 19 19oEb6 19/1 (29o19u1) 57/58 -30¢
28 zEb6 56/3 19or1 57/56 31¢
27 D6 18/1 18 D6 18/1
26 3oD6 52/3 17 17oDb6 17/1 (17u3o1) 51/52 -34¢
25 yyC#6 50/3 17ogg2 51/50 34¢
24 C6 16/1 16 C6 16/1
23 23oCb6 46/3 15 yB5 15/1 (23og2) 45/46 -38¢
22 1oB5 44/3 1uy1 45/44 39¢
21 zBb5 14/1 14 zBb5 14/1
20 yA5 40/3 13 3oA5 13/1 (3uy1) 39/40 -44¢
19 19oAb5 38/3 19u3o1 39/38 45¢
18 G5 12/1 12 G5 12/1
17 17oGb5 34/3 11 1oF#5 11/1 (17o1u2) 33/34 -52¢
16 F5 32/3 1o1 33/32 53¢
15 yE5 10/1 10 yE5 10/1
14 zEb5 28/3 9 D5 9/1 (z2) 27/28 -63¢
13 3oD5 26/3 3u1 27/26 65¢
12 C5 8/1 8 C5 8/1
11 1oB4 22/3 7 zBb4 7/1 (1or1) 21/22 -80¢
10 yA4 20/3 zg2 21/20 84¢
9 G4 6/1 6 G4 6/1
8 F4 16/3 5 yE4 5/1 (g2) 15/16 -112¢
7 zEb4 14/3 ry1 15/14 119¢
6 C4 4/1 4 C4 4/1
5 yA3 10/3 3 G3 3/1 (y2) 9/10 -182¢
4 F3 8/3 w2 9/8 204¢
3 C3 2/1 2 C3 2/1
2 F2 4/3 1 C2 1/1 (w4) 3/4 -498¢
1 F1 2/3 w5 3/2 702¢
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