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42 zeta peak index (abbreviated 42zpi), is the equal-step tuning system obtained from the 42nd peak of the Riemann zeta function.
| Tuning
|
Strength
|
Closest edo
|
Integer limit
|
| ZPI
|
Steps
per 8ve
|
Step size
(cents)
|
Height
|
Integral
|
Gap
|
Edo
|
Octave (cents)
|
Consistent
|
Distinct
|
| Size
|
Stretch
|
| 42zpi
|
13.900253
|
86.329367
|
4.592177
|
0.984037
|
14.097244
|
14edo
|
1208.611136
|
8.611136
|
7
|
5
|
Theory
42zpi is the closest zeta peak index to 14edo. It is effectively a stretched version of 14edo which improves on its tuning of most primes.
Harmonic series
Approximation of harmonics in 42zpi
| Harmonic
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
15
|
16
|
| Error
|
Absolute (¢)
|
+8.6
|
-2.7
|
+17.2
|
-23.8
|
+5.9
|
-2.0
|
+25.8
|
-5.4
|
-15.2
|
-7.5
|
+14.5
|
-37.7
|
+6.6
|
-26.5
|
+34.4
|
| Relative (%)
|
+10.0
|
-3.1
|
+19.9
|
-27.5
|
+6.8
|
-2.3
|
+29.9
|
-6.3
|
-17.6
|
-8.7
|
+16.8
|
-43.7
|
+7.7
|
-30.7
|
+39.9
|
| Step
|
14
|
22
|
28
|
32
|
36
|
39
|
42
|
44
|
46
|
48
|
50
|
51
|
53
|
54
|
56
|
Approximation of harmonics in 42zpi
| Harmonic
|
17
|
18
|
19
|
20
|
21
|
22
|
23
|
24
|
25
|
26
|
27
|
28
|
29
|
30
|
31
|
32
|
| Error
|
Absolute (¢)
|
+15.8
|
+3.2
|
-4.1
|
-6.6
|
-4.7
|
+1.1
|
+10.5
|
+23.1
|
+38.8
|
-29.1
|
-8.1
|
+15.2
|
+40.8
|
-17.9
|
+11.7
|
+43.1
|
| Relative (%)
|
+18.3
|
+3.7
|
-4.7
|
-7.6
|
-5.4
|
+1.3
|
+12.1
|
+26.8
|
+44.9
|
-33.7
|
-9.4
|
+17.7
|
+47.3
|
-20.7
|
+13.5
|
+49.9
|
| Step
|
57
|
58
|
59
|
60
|
61
|
62
|
63
|
64
|
65
|
65
|
66
|
67
|
68
|
68
|
69
|
70
|
Scala file
! 42zpi.scl
! Created using Scale Workshop 3.1.0
!
! https://scaleworkshop.plainsound.org/scale/g8qoINpGf
!
42nd zeta peak index
14
!
86.329
172.659
258.988
345.317
431.647
517.976
604.306
690.635
776.964
863.294
949.623
1035.952
1122.282
1208.611
