71zpi: Difference between revisions

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!Cents
!Cents
!Ratios
!Ratios
! colspan="3" |[[Ups and Downs Notation|Ups and Downs Notation from 20EDO]]
! colspan="3" |[[Ups and Downs Notation]] from [[20edo|20EDO]]
! colspan="3" |[[Ups and Downs Notation]] from [[182edo|182EDO]]
|-
|-
|0
|0
|0.000
|0.000
|[[1/1]]
|[[1/1]]
|unison
|P1
|D
|unison
|unison
|P1
|P1
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|^1, ^m2
|^1, ^m2
|^D, ^Eb
|^D, ^Eb
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|
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|-
|-
|2
|2
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|^^1, ~2
|^^1, ~2
|^^D, vvE
|^^D, vvE
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|-
|-
|3
|3
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|vM2
|vM2
|vE
|vE
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|
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|-
|-
|4
|4
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|M2, m3
|M2, m3
|E, F
|E, F
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|-
|-
|5
|5
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|^m3
|^m3
|^F
|^F
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|-
|6
|6
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|~3
|~3
|^^F, vvF#
|^^F, vvF#
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|-
|-
|7
|7
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|vM3
|vM3
|vF#
|vF#
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|-
|-
|8
|8
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|M3, P4
|M3, P4
|F#, G
|F#, G
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|-
|-
|9
|9
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|^4
|^4
|^G
|^G
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|-
|-
|10
|10
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|~4, ~5
|~4, ~5
|^^G, vvA
|^^G, vvA
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|-
|-
|11
|11
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|v5
|v5
|vA
|vA
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|-
|-
|12
|12
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|P5, m6
|P5, m6
|A
|A
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|-
|-
|13
|13
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|^5, ^m6
|^5, ^m6
|^A, ^Bb
|^A, ^Bb
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|-
|-
|14
|14
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|~6
|~6
|^^A, vvB
|^^A, vvB
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|-
|-
|15
|15
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|vM6
|vM6
|vB
|vB
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|-
|16
|16
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|M6, m7
|M6, m7
|B, C
|B, C
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|-
|17
|17
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|^m7
|^m7
|^C
|^C
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|-
|18
|18
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|~7
|~7
|^^C, vvD
|^^C, vvD
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|-
|-
|19
|19
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|vM7
|vM7
|vD
|vD
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|-
|-
|20
|20
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|P8
|P8
|D
|D
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|-
|-
|22
|22
|1305.326
|1305.326
|[[17/8]]
|[[17/8]]
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|1364.659
|1364.659
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|1483.325
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|1601.990
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|1661.323
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|1898.655
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|1957.988
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|2017.321
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|2076.654
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|2135.987
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|2195.320
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Revision as of 10:48, 21 April 2024

71 zeta peak index (abbreviated 71zpi), is the equal-step tuning system obtained from the 71st peak of the Riemann zeta function.

Tuning Strength Closest EDO Integer limit
ZPI Steps per octave Step size (cents) Height Integral Gap EDO Octave (cents) Consistent Distinct
71zpi 20.2248393119540 59.3329806724710 3.531097 0.613581 12.986080 20edo 1186.65961344942 6 6
The Riemann zeta function around 71zpi

Theory

71zpi marks the most prominent zeta peak index in the vicinity of 20edo. While 70zpi is the nearest peak to 20edo and closely competes with 71zpi in terms of strength, 71zpi remains superior across all measures of strength.

71zpi features a good 3:5:9:11:14:15:16:19:25:26:33 chord, which differs a lot from the harmonic characteristics of 20edo.

The nearest zeta peaks to 71zpi that surpass its strength are 65zpi and 75zpi.

Harmonic series

Approximation of harmonics in 71zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) -13.3 -3.3 -26.7 +2.3 -16.6 +13.2 +19.3 -6.6 -11.0 +2.0 +29.4 +9.4 -0.2 -1.0 +6.0
Relative (%) -22.5 -5.6 -45.0 +3.9 -28.0 +22.2 +32.5 -11.1 -18.5 +3.4 +49.5 +15.9 -0.3 -1.6 +10.1
Step 20 32 40 47 52 57 61 64 67 70 73 75 77 79 81
Approximation of harmonics in 71zpi
Harmonic 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
Error Absolute (¢) +19.7 -19.9 +5.1 -24.3 +9.9 -11.3 -29.0 +16.0 +4.7 -3.9 -9.9 -13.5 -14.9 -14.3 -11.7 -7.4 -1.3
Relative (%) +33.2 -33.6 +8.6 -41.0 +16.6 -19.1 -48.8 +27.0 +7.9 -6.6 -16.7 -22.8 -25.2 -24.1 -19.8 -12.4 -2.2
Step 83 84 86 87 89 90 91 93 94 95 96 97 98 99 100 101 102

Intervals

The table bellow presents 32 integer-limit ratios by direct mapping to 71zpi steps, with an allowable error determined by the formula: abs (cents_error) < (1200 / (n * d)), where n/d represents the octave-reduced ratio. If you have an alternative or more effective formula, please feel free to suggest it.

There are multiple ways to approach notation. The simplest method is to use the notations from 20edo. However, this approach will not preserve octave compression when the audio is rendered by notation software. If maintaining accurate step compression in notation software is important, consider using the ups and downs notation from 182edo at every 9-degree. With this method, the tonal difference will be less than 1 cent up to the 86th harmonic.

Step Cents Ratios Ups and Downs Notation from 20EDO Ups and Downs Notation from 182EDO
0 0.000 1/1 unison P1 D unison P1 D
1 59.333 30/29, 29/28 up unison, upminor 2nd ^1, ^m2 ^D, ^Eb
2 118.666 15/14 dup unison, mid 2nd ^^1, ~2 ^^D, vvE
3 177.999 10/9 downmajor 2nd vM2 vE
4 237.332 8/7 major 2nd, minor 3rd M2, m3 E, F
5 296.665 13/11, 19/16, 6/5 upminor 3rd ^m3 ^F
6 355.998 11/9, 27/22, 16/13 mid 3rd ~3 ^^F, vvF#
7 415.331 5/4, 14/11 downmajor 3rd vM3 vF#
8 474.664 25/19, 4/3 major 3rd, perfect fourth M3, P4 F#, G
9 533.997 15/11 up-fourth ^4 ^G
10 593.330 7/5, 31/22 mid fourth, mid fifth ~4, ~5 ^^G, vvA
11 652.663 16/11, 19/13 down-fifth v5 vA
12 711.996 3/2 fifth P5, m6 A
13 771.329 14/9, 25/16, 11/7 upfifth, upminor 6th ^5, ^m6 ^A, ^Bb
14 830.662 8/5, 21/13, 13/8 mid 6th ~6 ^^A, vvB
15 889.995 5/3 downmajor 6th vM6 vB
16 949.328 19/11, 26/15, 7/4 major 6th, minor 7th M6, m7 B, C
17 1008.661 9/5 upminor 7th ^m7 ^C
18 1067.994 13/7 mid 7th ~7 ^^C, vvD
19 1127.327 23/12 downmajor 7th vM7 vD
20 1186.660 2/1 octave P8 D
22 1305.326 17/8
23 1364.659 11/5
25 1483.325 7/3
27 1601.990 5/2
28 1661.323 13/5
29 1720.656 8/3, 27/10
30 1779.989 14/5
32 1898.655 3/1
33 1957.988 31/10
34 2017.321 16/5
35 2076.654 10/3
36 2135.987 24/7
37 2195.320 7/2, 32/9
38 2254.653 11/3
39 2313.986 19/5
40 2373.319 4/1
44 2610.651 9/2
45 2669.984 14/3
46 2729.317 29/6
47 2788.650 5/1
51 3025.982 23/4
52 3085.315 6/1
57 3381.980 7/1
61 3619.312 8/1
63 3737.978 26/3
64 3797.311 9/1
67 3975.310 10/1
70 4153.309 11/1
75 4449.974 13/1
77 4568.640 14/1
78 4627.972 29/2
79 4687.305 15/1
80 4746.638 31/2
81 4805.971 16/1

Approximation to JI

Intervals by direct approximation (even if inconsistent)
Ratio Error (abs, ¢) Error (rel, %)
14/1 0.186 0.314
11/5 0.346 0.583
17/8 0.370 0.624
31/22 0.388 0.654
21/13 0.408 0.688
25/19 0.451 0.759
26/3 0.595 1.003
30/29 0.641 1.081
31/10 0.733 1.236
32/9 0.770 1.297
15/14 0.777 1.309
19/16 0.848 1.429
15/1 0.963 1.623
23/12 1.007 1.698
27/10 1.105 1.863
25/16 1.299 2.189
29/28 1.418 2.390
27/22 1.451 2.445
31/2 1.603 2.702
29/2 1.605 2.705
29/6 1.695 2.857
11/1 1.991 3.355
14/11 2.177 3.669
23/4 2.292 3.864
5/1 2.336 3.938
14/5 2.523 4.252
19/5 2.787 4.697
24/7 2.858 4.817
26/15 2.931 4.940
15/11 2.954 4.979
14/3 3.113 5.247
19/11 3.133 5.280
3/1 3.300 5.561
16/13 3.474 5.856
16/5 3.635 6.127
13/7 3.708 6.250
16/11 3.981 6.709
19/13 4.323 7.285
10/9 4.405 7.424
11/3 5.290 8.916
5/3 5.636 9.499
16/1 5.971 10.064
8/7 6.158 10.378
14/9 6.413 10.808
9/1 6.599 11.122
9/2 6.741 11.362
13/5 7.110 11.982
13/11 7.455 12.565
10/3 7.704 12.985
11/9 8.590 14.478
9/5 8.936 15.060
13/1 9.446 15.920
13/8 9.866 16.628
3/2 10.041 16.923
7/5 10.818 18.232
10/1 11.004 18.546
11/7 11.163 18.815
7/1 13.154 22.170
2/1 13.340 22.484
5/2 15.677 26.422
7/3 16.454 27.731
6/1 16.640 28.045
8/5 16.975 28.610
6/5 18.976 31.983
8/1 19.312 32.548
7/4 19.498 32.862
8/3 22.611 38.109
4/3 23.381 39.407
7/2 26.494 44.654
4/1 26.681 44.968
5/4 29.017 48.906
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