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'''71 zeta peak index''' (abbreviated '''71zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 71st peak of the [[The Riemann zeta function and tuning|Riemann zeta function]].
'''71 zeta peak index''' (abbreviated '''71zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 71st [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].


{|class="wikitable"
{{ZPI
!colspan="3"|Tuning
| zpi = 71
!colspan="3"|Strength
| steps = 20.2248393119540
!colspan="2"|Closest EDO
| step size = 59.3329806724710
!colspan="2"|Integer limit
| height = 3.531097
| integral = 0.613581
| gap = 12.986080
| octave = 1186.65961344942
| consistent = 6
| distinct = 6
}}
 
[[File:71zpi.png|thumb|right|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 may also be viewed as a tritave compression of [[32edt]], a [[The_Riemann_zeta_function_and_tuning#Removing_primes|no-2s zeta peak EDT]] (consistent in the [[Odd_limit#Nonoctave_equaves|no-2s 19-integer-limit]]), but with less extreme stretch than [[71zpi#Record on the Riemann zeta function with prime 2 removed|the no-2s peak]] at 59.271105 cents.
 
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]].
 
71zpi is distinguished by its extensive [[EDO-span|EDO-deviation]] and substantial zeta strength, qualifying it as a strong candidate for no-octave tuning systems. It is noteworthy that only [[19zpi]] exhibits both a greater octave error and stronger zeta height and integral than 71zpi, although 71zpi still has a more pronounced zeta gap. Other notable [[Zeta peak index|zeta peak indices]] in this category include [[61zpi]], [[84zpi]], [[110zpi]], [[137zpi]], [[151zpi]], [[222zpi]], and [[273zpi]], each demonstrating characteristics that make them suitable for similar applications.
 
=== Harmonic series ===
{{Harmonics in cet|59.3329806724710|columns=15|title=Approximation of harmonics in 71zpi}}
{{Harmonics in cet|59.3329806724710|columns=17|start=16|title=Approximation of harmonics in 71zpi}}
 
== Intervals ==
 
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 step. With this method, the tonal difference will be less than 1 cent up to the 86th harmonic.
 
{| class="wikitable center-1 right-2 left-3 center-4 center-5"
|+ style="white-space:nowrap" | Intervals in 71zpi
|-
| colspan="3" style="text-align:left;" | JI ratios are comprised of 33-integer-limit ratios,<br>and are stylized as follows to indicate their accuracy:
* '''<u>Bold Underlined:</u>''' relative error < 8.333 %
* '''Bold:''' relative error < 16.667 %
* Normal: relative error < 25 %
* <small>Small:</small> relative error < 33.333 %
* <small><small>Small Small:</small></small> relative error < 41.667 %
* <small><small><small>Small Small Small:</small></small></small> relative error < 50 %
| colspan="2" style="text-align:right;" | <center>'''⟨182 288] at every 9 steps'''</center><br>[[9/8|Whole tone]] = 30 steps<br>[[256/243|Limma]] = 16 steps<br>[[2187/2048|Apotome]] = 14 steps
|-
! Degree
! Cents
! Ratios
! Ups and downs notation
! Step
|-
| 0
| 0.000
|
| P1
| 0
|-
| 1
| 59.333
| '''[[33/32]]''', '''<u>[[32/31]]'''</u>, '''<u>[[31/30]]'''</u>, '''<u>[[30/29]]'''</u>, '''<u>[[29/28]]'''</u>, '''<u>[[28/27]]'''</u>, '''[[27/26]]''', '''[[26/25]]''', [[25/24]], [[24/23]], <small>[[23/22]]</small>, <small><small>[[22/21]]</small></small>, <small><small><small>[[21/20]]</small></small></small>, <small><small><small>[[20/19]]</small></small></small>
| v<sup>7</sup>m2
| 9
|-
| 2
| 118.666
| <small><small><small>[[19/18]]</small></small></small>, <small>[[18/17]]</small>, [[17/16]], [[33/31]], '''[[16/15]]''', '''<u>[[31/29]]'''</u>, '''<u>[[15/14]]'''</u>, '''[[29/27]]''', '''[[14/13]]''', [[27/25]], <small><small>[[13/12]]</small></small>, <small><small><small>[[25/23]]</small></small></small>
| ^^m2
| 18
|-
| 3
| 177.999
| <small><small><small>[[12/11]]</small></small></small>, <small><small>[[23/21]]</small></small>, [[11/10]], '''[[32/29]]''', '''<u>[[21/19]]'''</u>, '''<u>[[31/28]]'''</u>, '''<u>[[10/9]]'''</u>, [[29/26]], [[19/17]], <small>[[28/25]]</small>, <small><small><small>[[9/8]]</small></small></small>
| vvvM2
| 27
|-
| 4
| 237.332
| <small><small><small>[[26/23]]</small></small></small>, <small><small>[[17/15]]</small></small>, <small>[[25/22]]</small>, [[33/29]], '''[[8/7]]''', '''<u>[[31/27]]'''</u>, '''<u>[[23/20]]'''</u>, [[15/13]], <small>[[22/19]]</small>, <small>[[29/25]]</small>, <small><small><small>[[7/6]]</small></small></small>
| ^<sup>6</sup>M2
| 36
|-
| 5
| 296.665
| <small>[[27/23]]</small>, <small>[[20/17]]</small>, [[33/28]], '''[[13/11]]''', '''<u>[[32/27]]'''</u>, '''<u>[[19/16]]'''</u>, '''[[25/21]]''', '''[[31/26]]''', <small>[[6/5]]</small>
| vm3
| 45
|-
| 6
| 355.998
| <small><small><small>[[29/24]]</small></small></small>, <small><small><small>[[23/19]]</small></small></small>, <small><small>[[17/14]]</small></small>, <small>[[28/23]]</small>, '''[[11/9]]''', '''<u>[[27/22]]'''</u>, '''<u>[[16/13]]'''</u>, '''[[21/17]]''', [[26/21]], <small>[[31/25]]</small>
| v<sup>6</sup>M3
| 54
|-
| 7
| 415.331
| <small><small><small>[[5/4]]</small></small></small>, [[29/23]], [[24/19]], '''[[19/15]]''', '''<u>[[33/26]]'''</u>, '''<u>[[14/11]]'''</u>, '''[[23/18]]''', [[32/25]], <small>[[9/7]]</small>, <small><small><small>[[31/24]]</small></small></small>
| ^^^M3
| 63
|-
| 8
| 474.664
| <small><small><small>[[22/17]]</small></small></small>, <small><small>[[13/10]]</small></small>, [[30/23]], [[17/13]], '''<u>[[21/16]]'''</u>, '''<u>[[25/19]]'''</u>, '''<u>[[29/22]]'''</u>, '''[[33/25]]''', <small><small>[[4/3]]</small></small>
| v<sup>4</sup>4
| 72
|-
| 9
| 533.997
| <small>[[31/23]]</small>, [[27/20]], [[23/17]], '''[[19/14]]''', '''<u>[[15/11]]'''</u>, '''[[26/19]]''', <small>[[11/8]]</small>, <small><small><small>[[29/21]]</small></small></small>, <small><small><small>[[18/13]]</small></small></small>
| ^<sup>5</sup>4
| 81
|-
| 10
| 593.330
| <small><small>[[25/18]]</small></small>, <small><small>[[32/23]]</small></small>, [[7/5]], '''<u>[[31/22]]'''</u>, '''<u>[[24/17]]'''</u>, '''[[17/12]]''', <small>[[27/19]]</small>, <small><small>[[10/7]]</small></small>
| A4
| 90
|-
| 11
| 652.663
| <small><small><small>[[33/23]]</small></small></small>, <small><small>[[23/16]]</small></small>, <small>[[13/9]]</small>, '''[[29/20]]''', '''<u>[[16/11]]'''</u>, '''<u>[[19/13]]'''</u>, [[22/15]], <small>[[25/17]]</small>, <small>[[28/19]]</small>, <small><small>[[31/21]]</small></small>
| ~5
| 99
|-
| 12
| 711.996
| [[3/2]], <small>[[32/21]]</small>, <small><small>[[29/19]]</small></small>, <small><small>[[26/17]]</small></small>, <small><small><small>[[23/15]]</small></small></small>
| ^^5
| 108
|-
| 13
| 771.329
| <small><small><small>[[20/13]]</small></small></small>, <small>[[17/11]]</small>, [[31/20]], '''[[14/9]]''', '''<u>[[25/16]]'''</u>, [[11/7]], <small>[[30/19]]</small>, <small><small>[[19/12]]</small></small>, <small><small><small>[[27/17]]</small></small></small>
| v<sup>5</sup>m6
| 117
|-
| 14
| 830.662
| <small>[[8/5]]</small>, '''[[29/18]]''', '''<u>[[21/13]]'''</u>, '''[[13/8]]''', <small>[[31/19]]</small>, <small><small>[[18/11]]</small></small>, <small><small><small>[[23/14]]</small></small></small>
| ^<sup>4</sup>m6
| 126
|-
| 15
| 889.995
| <small><small><small>[[28/17]]</small></small></small>, <small><small>[[33/20]]</small></small>, '''[[5/3]]''', [[32/19]], <small>[[27/16]]</small>, <small><small>[[22/13]]</small></small>, <small><small><small>[[17/10]]</small></small></small>
| vM6
| 135
|-
| 16
| 949.328
| <small><small>[[29/17]]</small></small>, <small>[[12/7]]</small>, '''[[31/18]]''', '''<u>[[19/11]]'''</u>, '''<u>[[26/15]]'''</u>, '''[[33/19]]''', <small>[[7/4]]</small>
| v<sup>6</sup>A6, ^<sup>6</sup>d7
| 144
|-
| 17
| 1008.661
| <small><small><small>[[30/17]]</small></small></small>, <small><small>[[23/13]]</small></small>, [[16/9]], '''<u>[[25/14]]'''</u>, '''[[9/5]]''', <small><small>[[29/16]]</small></small>, <small><small><small>[[20/11]]</small></small></small>
| ^m7
| 153
|-
| 18
| 1067.994
| <small><small><small>[[31/17]]</small></small></small>, <small>[[11/6]]</small>, '''[[24/13]]''', '''<u>[[13/7]]'''</u>, [[28/15]], <small><small>[[15/8]]</small></small>, <small><small><small>[[32/17]]</small></small></small>
| v<sup>4</sup>M7
| 162
|-
| 19
| 1127.327
| <small><small><small>[[17/9]]</small></small></small>, <small>[[19/10]]</small>, '''[[21/11]]''', '''<u>[[23/12]]'''</u>, '''<u>[[25/13]]'''</u>, '''[[27/14]]''', [[29/15]], <small>[[31/16]]</small>, <small><small>[[33/17]]</small></small>
| ^<sup>5</sup>M7
| 171
|-
| 20
| 1186.660
| [[2/1]]
| vv1 +1 oct
| 180
|-
| 21
| 1245.993
| '''[[33/16]]''', [[31/15]], [[29/14]], <small>[[27/13]]</small>, <small><small>[[25/12]]</small></small>
| ^<sup>7</sup>1 +1 oct
| 189
|-
| 22
| 1305.326
| <small><small><small>[[23/11]]</small></small></small>, <small><small>[[21/10]]</small></small>, [[19/9]], '''<u>[[17/8]]'''</u>, '''[[32/15]]''', [[15/7]], <small><small>[[28/13]]</small></small>
| m2 +1 oct
| 198
|-
| 23
| 1364.659
| <small><small><small>[[13/6]]</small></small></small>, [[24/11]], '''<u>[[11/5]]'''</u>, [[31/14]], <small>[[20/9]]</small>, <small><small>[[29/13]]</small></small>
| v<sup>5</sup>M2 +1 oct
| 207
|-
| 24
| 1423.992
| <small><small>[[9/4]]</small></small>, '''<u>[[25/11]]'''</u>, '''[[16/7]]''', <small>[[23/10]]</small>, <small><small>[[30/13]]</small></small>
| ^<sup>4</sup>M2 +1 oct
| 216
|-
| 25
| 1483.325
| <small>[[7/3]]</small>, '''<u>[[33/14]]'''</u>, '''[[26/11]]''', [[19/8]], <small><small>[[31/13]]</small></small>
| vvvm3 +1 oct
| 225
|-
| 26
| 1542.657
| <small><small><small>[[12/5]]</small></small></small>, <small>[[29/12]]</small>, '''[[17/7]]''', '''<u>[[22/9]]'''</u>, [[27/11]], <small>[[32/13]]</small>
| ^<sup>6</sup>m3 +1 oct
| 234
|-
| 27
| 1601.990
| <small>[[5/2]]</small>, [[33/13]], <small>[[28/11]]</small>, <small><small>[[23/9]]</small></small>
| ^M3 +1 oct
| 243
|-
| 28
| 1661.323
| <small><small><small>[[18/7]]</small></small></small>, <small>[[31/12]]</small>, '''[[13/5]]''', '''[[21/8]]''', <small>[[29/11]]</small>
| v<sup>6</sup>4 +1 oct
| 252
|-
| 29
| 1720.656
| <small><small>[[8/3]]</small></small>, '''<u>[[27/10]]'''</u>, '''[[19/7]]''', <small>[[30/11]]</small>
| ^^^4 +1 oct
| 261
|-
| 30
| 1779.989
| <small><small><small>[[11/4]]</small></small></small>, [[25/9]], '''<u>[[14/5]]'''</u>, [[31/11]], <small><small>[[17/6]]</small></small>
| vvA4 +1 oct
| 270
|-
| 31
| 1839.322
| <small><small>[[20/7]]</small></small>, [[23/8]], '''<u>[[26/9]]'''</u>, '''<u>[[29/10]]'''</u>, '''[[32/11]]'''
| ^<sup>5</sup>d5 +1 oct
| 279
|-
| 32
| 1898.655
| '''<u>[[3/1]]'''</u>
| P5 +1 oct
| 288
|-
| 33
| 1957.988
| '''<u>[[31/10]]'''</u>, '''[[28/9]]''', [[25/8]], <small><small>[[22/7]]</small></small>
| v<sup>7</sup>m6 +1 oct
| 297
|-
| 34
| 2017.321
| <small><small>[[19/6]]</small></small>, '''<u>[[16/5]]'''</u>, '''[[29/9]]''', <small><small>[[13/4]]</small></small>
| ^^m6 +1 oct
| 306
|-
| 35
| 2076.654
| <small>[[23/7]]</small>, '''[[33/10]]''', '''[[10/3]]''', <small><small><small>[[27/8]]</small></small></small>
| vvvM6 +1 oct
| 315
|-
| 36
| 2135.987
| <small>[[17/5]]</small>, '''<u>[[24/7]]'''</u>, '''[[31/9]]'''
| ^<sup>6</sup>M6 +1 oct
| 324
|-
| 37
| 2195.320
| <small><small><small>[[7/2]]</small></small></small>, '''<u>[[32/9]]'''</u>, '''[[25/7]]''', <small><small>[[18/5]]</small></small>
| vm7 +1 oct
| 333
|-
| 38
| 2254.653
| <small><small><small>[[29/8]]</small></small></small>, '''[[11/3]]''', <small>[[26/7]]</small>
| v<sup>6</sup>M7 +1 oct
| 342
|-
| 39
| 2313.986
| <small><small><small>[[15/4]]</small></small></small>, '''<u>[[19/5]]'''</u>, [[23/6]], <small><small>[[27/7]]</small></small>
| ^^^M7 +1 oct
| 351
|-
| 40
| 2373.319
| <small><small><small>[[31/8]]</small></small></small>, <small><small><small>[[4/1]]</small></small></small>
| v<sup>4</sup>1 +2 oct
| 360
|-
| 41
| 2432.652
| <small><small>[[33/8]]</small></small>, <small><small><small>[[29/7]]</small></small></small>
| ^<sup>5</sup>1 +2 oct
| 369
|-
| 42
| 2491.985
| <small><small>[[25/6]]</small></small>, '''[[21/5]]''', [[17/4]], <small><small><small>[[30/7]]</small></small></small>
| vvm2 +2 oct
| 378
|-
| 43
| 2551.318
| [[13/3]], [[22/5]], <small><small><small>[[31/7]]</small></small></small>
| ~2 +2 oct
| 387
|-
| 44
| 2610.651
| '''[[9/2]]''', <small><small>[[32/7]]</small></small>
| ^^M2 +2 oct
| 396
|-
| 45
| 2669.984
| <small><small><small>[[23/5]]</small></small></small>, '''<u>[[14/3]]'''</u>, [[33/7]], <small><small><small>[[19/4]]</small></small></small>
| v<sup>5</sup>m3 +2 oct
| 405
|-
| 46
| 2729.317
| [[24/5]], '''<u>[[29/6]]'''</u>
| ^<sup>4</sup>m3 +2 oct
| 414
|-
| 47
| 2788.650
| '''<u>[[5/1]]'''</u>
| vM3 +2 oct
| 423
|-
| 48
| 2847.983
| '''<u>[[31/6]]'''</u>, '''[[26/5]]''', <small><small>[[21/4]]</small></small>
| v<sup>6</sup>A3 +2 oct, ^<sup>6</sup>d4 +2 oct
| 432
|-
| 49
| 2907.316
| '''[[16/3]]''', [[27/5]]
| ^4 +2 oct
| 441
|-
| 50
| 2966.649
| <small>[[11/2]]</small>, <small>[[28/5]]</small>
| v<sup>4</sup>A4 +2 oct
| 450
|-
| 51
| 3025.982
| <small><small>[[17/3]]</small></small>, '''<u>[[23/4]]'''</u>, <small>[[29/5]]</small>
| ^^^d5 +2 oct
| 459
|-
| 52
| 3085.315
| <small>[[6/1]]</small>
| vv5 +2 oct
| 468
|-
| 53
| 3144.648
| [[31/5]], <small><small><small>[[25/4]]</small></small></small>
| ^<sup>7</sup>5 +2 oct
| 477
|-
| 54
| 3203.981
| '''[[19/3]]''', '''[[32/5]]'''
| m6 +2 oct
| 486
|-
| 55
| 3263.314
| <small><small>[[13/2]]</small></small>, '''<u>[[33/5]]'''</u>, <small><small>[[20/3]]</small></small>
| v<sup>5</sup>M6 +2 oct
| 495
|-
| 56
| 3322.647
| <small>[[27/4]]</small>
| ^<sup>4</sup>M6 +2 oct
| 504
|-
| 57
| 3381.980
| [[7/1]]
| vvvm7 +2 oct
| 513
|-
| 58
| 3441.313
| [[29/4]], '''[[22/3]]'''
| ^<sup>6</sup>m7 +2 oct
| 522
|-
| 59
| 3500.646
| [[15/2]], <small><small><small>[[23/3]]</small></small></small>
| ^M7 +2 oct
| 531
|-
| 60
| 3559.979
| <small>[[31/4]]</small>
| v<sup>6</sup>1 +3 oct
| 540
|-
| 61
| 3619.312
| <small>[[8/1]]</small>
| ^^^1 +3 oct
| 549
|-
| 62
| 3678.645
| <small><small><small>[[33/4]]</small></small></small>, '''[[25/3]]''', <small><small><small>[[17/2]]</small></small></small>
| v<sup>4</sup>m2 +3 oct
| 558
|-
| 63
| 3737.978
| '''<u>[[26/3]]'''</u>
| ^<sup>5</sup>m2 +3 oct
| 567
|-
| 64
| 3797.311
| '''[[9/1]]'''
| M2 +3 oct
| 576
|-
| 65
| 3856.644
| [[28/3]]
| v<sup>7</sup>m3 +3 oct
| 585
|-
| 66
| 3915.977
| <small>[[19/2]]</small>, [[29/3]]
| ^^m3 +3 oct
| 594
|-
| 67
| 3975.310
| [[10/1]]
| vvvM3 +3 oct
| 603
|-
| 68
| 4034.643
| '''[[31/3]]'''
| ^<sup>6</sup>M3 +3 oct
| 612
|-
| 69
| 4093.976
| <small><small>[[21/2]]</small></small>, '''<u>[[32/3]]'''</u>
| v4 +3 oct
| 621
|-
| 70
| 4153.309
| '''<u>[[11/1]]'''</u>
| v<sup>6</sup>A4 +3 oct
| 630
|-
| 71
| 4212.642
| <small>[[23/2]]</small>
| ^d5 +3 oct
| 639
|-
| 72
| 4271.975
|
| v<sup>4</sup>5 +3 oct
| 648
|-
| 73
| 4331.308
| <small><small><small>[[12/1]]</small></small></small>
| ^<sup>5</sup>5 +3 oct
| 657
|-
| 74
| 4390.641
| <small>[[25/2]]</small>
| vvm6 +3 oct
| 666
|-
| 75
| 4449.974
| '''[[13/1]]'''
| ~6 +3 oct
| 675
|-
| 76
| 4509.307
| '''<u>[[27/2]]'''</u>
| ^^M6 +3 oct
| 684
|-
| 77
| 4568.640
| '''<u>[[14/1]]'''</u>
| v<sup>5</sup>m7 +3 oct
| 693
|-
| 78
| 4627.972
| '''<u>[[29/2]]'''</u>
| ^<sup>4</sup>m7 +3 oct
| 702
|-
| 79
| 4687.305
| '''<u>[[15/1]]'''</u>
| vM7 +3 oct
| 711
|-
| 80
| 4746.638
| '''<u>[[31/2]]'''</u>
| v<sup>6</sup>A7 +3 oct, ^<sup>6</sup>d1 +4 oct
| 720
|-
| 81
| 4805.971
| '''[[16/1]]'''
| ^1 +4 oct
| 729
|-
| 82
| 4865.304
| [[33/2]]
| v<sup>6</sup>m2 +4 oct
| 738
|-
| 83
| 4924.637
| <small>[[17/1]]</small>
| ^^^m2 +4 oct
| 747
|-
| 84
| 4983.970
| <small><small>[[18/1]]</small></small>
| vvM2 +4 oct
| 756
|-
| 85
| 5043.303
|
| ^<sup>7</sup>M2 +4 oct
| 765
|-
| 86
| 5102.636
| '''[[19/1]]'''
| m3 +4 oct
| 774
|-
| 87
| 5161.969
| <small><small>[[20/1]]</small></small>
| v<sup>5</sup>M3 +4 oct
| 783
|-
| 88
| 5221.302
|
| ^<sup>4</sup>M3 +4 oct
| 792
|-
| 89
| 5280.635
| '''[[21/1]]'''
| vvv4 +4 oct
| 801
|-
| 90
| 5339.968
| [[22/1]]
| ^<sup>6</sup>4 +4 oct
| 810
|-
| 91
| 5399.301
| <small><small><small>[[23/1]]</small></small></small>
| ^A4 +4 oct, vd5 +4 oct
| 819
|-
| 92
| 5458.634
|
| v<sup>6</sup>5 +4 oct
| 828
|-
| 93
| 5517.967
| <small>[[24/1]]</small>
| ^^^5 +4 oct
| 837
|-
| 94
| 5577.300
| '''<u>[[25/1]]'''</u>
| v<sup>4</sup>m6 +4 oct
| 846
|-
| 95
| 5636.633
| '''<u>[[26/1]]'''</u>
| ^<sup>5</sup>m6 +4 oct
| 855
|-
| 96
| 5695.966
| [[27/1]]
| M6 +4 oct
| 864
|-
| 97
| 5755.299
| [[28/1]]
| v<sup>7</sup>m7 +4 oct
| 873
|-
| 98
| 5814.632
| <small>[[29/1]]</small>
| ^^m7 +4 oct
| 882
|-
| 99
| 5873.965
| [[30/1]]
| vvvM7 +4 oct
| 891
|-
| 100
| 5933.298
| [[31/1]]
| ^<sup>6</sup>M7 +4 oct
| 900
|-
| 101
| 5992.631
| '''[[32/1]]'''
| v1 +5 oct
| 909
|-
| 102
| 6051.964
| '''<u>[[33/1]]'''</u>
| v<sup>6</sup>A1 +5 oct, ^<sup>6</sup>d2 +5 oct
| 918
|}
 
== Approximation to JI ==
 
=== Interval mappings ===
 
The following tables show how 33-integer-limit intervals are represented in 71zpi. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italics''.
 
{| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 33-integer-limit intervals in 71zpi (by direct approximation)
|-
! Ratio
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
| [[14/1]]
| -0.186
| -0.314
|-
| [[11/5]]
| -0.346
| -0.583
|- style="background-color: #cccccc;"
| ''[[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
|- style="background-color: #cccccc;"
| ''[[32/9]]''
| ''-0.770''
| ''-1.297''
|-
| [[15/14]]
| -0.777
| -1.309
|- style="background-color: #cccccc;"
| ''[[19/16]]''
| ''-0.848''
| ''-1.429''
|-
| [[15/1]]
| -0.963
| -1.623
|-
| [[23/12]]
| +1.007
| +1.698
|-
| [[27/10]]
| +1.105
| +1.863
|-
| [[33/14]]
| -1.123
| -1.892
|- style="background-color: #cccccc;"
| ''[[25/16]]''
| ''-1.299''
| ''-2.189''
|-
| [[33/1]]
| -1.309
| -2.206
|-
| [[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
|-
| [[31/28]]
| +1.789
| +3.016
|-
| [[31/27]]
| -1.839
| -3.099
|-
| '''[[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
|- style="background-color: #cccccc;"
| ''[[32/27]]''
| ''+2.530''
| ''+4.264''
|-
| [[31/30]]
| +2.566
| +4.325
|-
| [[33/26]]
| +2.586
| +4.358
|-
| [[25/11]]
| +2.682
| +4.520
|-
| [[26/9]]
| +2.705
| +4.559
|-
| [[19/5]]
| +2.787
| +4.697
|- style="background-color: #cccccc;"
| ''[[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
|-
| [[31/29]]
| +3.208
| +5.406
|-
| '''[[3/1]]'''
| '''-3.300'''
| '''-5.561'''
|-
| [[27/2]]
| +3.442
| +5.800
|- style="background-color: #cccccc;"
| ''[[16/13]]''
| ''-3.474''
| ''-5.856''
|-
| [[29/22]]
| -3.595
| -6.060
|-
| [[28/27]]
| -3.628
| -6.115
|- style="background-color: #cccccc;"
| ''[[16/5]]''
| ''+3.635''
| ''+6.127''
|-
| [[33/5]]
| -3.645
| -6.144
|- style="background-color: #cccccc;"
| ''[[24/17]]''
| ''-3.670''
| ''-6.185''
|-
| [[13/7]]
| -3.708
| -6.250
|- style="background-color: #cccccc;"
| ''[[21/16]]''
| ''+3.883''
| ''+6.544''
|-
| [[26/1]]
| -3.894
| -6.564
|-
| [[29/10]]
| -3.941
| -6.642
|- style="background-color: #cccccc;"
| ''[[16/11]]''
| ''+3.981''
| ''+6.709''
|- style="background-color: #cccccc;"
| ''[[32/3]]''
| ''-4.069''
| ''-6.858''
|-
| [[19/13]]
| -4.323
| -7.285
|- style="background-color: #cccccc;"
| ''[[32/31]]''
| ''+4.369''
| ''+7.363''
|-
| [[10/9]]
| -4.405
| -7.424
|-
| [[23/20]]
| -4.629
| -7.801
|-
| [[25/1]]
| +4.673
| +7.875
|-
| [[21/19]]
| +4.731
| +7.974
|-
| [[22/9]]
| -4.750
| -8.006
|-
| [[25/13]]
| -4.773
| -8.045
|-
| [[25/14]]
| +4.859
| +8.190
|-
| [[31/6]]
| +4.903
| +8.263
|-
| [[29/18]]
| +4.995
| +8.418
|-
| [[29/27]]
| -5.046
| -8.505
|-
| '''[[19/1]]'''
| '''+5.123'''
| '''+8.635'''
|-
| [[31/9]]
| -5.138
| -8.660
|-
| [[25/21]]
| -5.182
| -8.733
|-
| [[11/3]]
| +5.290
| +8.916
|-
| [[19/14]]
| +5.310
| +8.949
|-
| [[5/3]]
| +5.636
| +9.499
|-
| [[26/11]]
| -5.885
| -9.919
|- style="background-color: #cccccc;"
| ''[[16/1]]''
| ''+5.971''
| ''+10.064''
|-
| [[33/25]]
| -5.982
| -10.082
|-
| [[27/26]]
| -6.004
| -10.120
|- style="background-color: #cccccc;"
| ''[[33/32]]''
| ''+6.060''
| ''+10.214''
|-
| [[19/15]]
| +6.087
| +10.258
|- style="background-color: #cccccc;"
| ''[[8/7]]''
| ''+6.158''
| ''+10.378''
|-
| [[26/5]]
| -6.231
| -10.502
|- style="background-color: #cccccc;"
| ''[[32/15]]''
| ''-6.406''
| ''-10.796''
|-
| [[14/9]]
| +6.413
| +10.808
|-
| [[33/19]]
| -6.432
| -10.841
|-
| [[17/7]]
| +6.528
| +11.002
|- style="background-color: #cccccc;"
| ''[[24/13]]''
| ''+6.566''
| ''+11.067''
|-
| [[9/1]]
| -6.599
| -11.122
|-
| [[9/2]]
| +6.741
| +11.362
|-
| [[28/9]]
| -6.928
| -11.676
|- style="background-color: #cccccc;"
| ''[[16/15]]''
| ''+6.935''
| ''+11.688''
|-
| [[13/5]]
| +7.110
| +11.982
|- style="background-color: #cccccc;"
| ''[[16/7]]''
| ''-7.183''
| ''-12.106''
|- style="background-color: #cccccc;"
| ''[[33/16]]''
| ''-7.280''
| ''-12.270''
|- style="background-color: #cccccc;"
| ''[[32/1]]''
| ''-7.369''
| ''-12.420''
|-
| [[13/11]]
| +7.455
| +12.565
|-
| [[21/5]]
| +7.518
| +12.671
|- style="background-color: #cccccc;"
| ''[[32/29]]''
| ''+7.576''
| ''+12.769''
|-
| [[10/3]]
| -7.704
| -12.985
|-
| [[31/26]]
| -7.843
| -13.219
|-
| [[21/11]]
| +7.864
| +13.253
|-
| [[25/3]]
| +7.972
| +13.437
|-
| [[19/7]]
| -8.031
| -13.535
|-
| [[22/3]]
| -8.050
| -13.568
|-
| [[31/18]]
| +8.202
| +13.824
|-
| [[29/9]]
| -8.346
| -14.066
|-
| [[19/3]]
| +8.423
| +14.196
|-
| [[31/3]]
| -8.438
| -14.221
|-
| [[25/7]]
| -8.481
| -14.294
|-
| [[26/25]]
| -8.567
| -14.439
|-
| [[11/9]]
| +8.590
| +14.478
|-
| [[9/5]]
| -8.936
| -15.060
|-
| [[26/19]]
| -9.018
| -15.199
|-
| [[23/18]]
| -9.033
| -15.225
|- style="background-color: #cccccc;"
| ''[[16/3]]''
| ''+9.271''
| ''+15.625''
|- style="background-color: #cccccc;"
| ''[[32/11]]''
| ''-9.360''
| ''-15.775''
|-
| [[29/20]]
| +9.399
| +15.842
|-
| '''[[13/1]]'''
| '''+9.446'''
| '''+15.920'''
|- style="background-color: #cccccc;"
| ''[[21/8]]''
| ''-9.457''
| ''-15.940''
|-
| [[14/13]]
| -9.632
| -16.234
|- style="background-color: #cccccc;"
| ''[[17/12]]''
| ''-9.671''
| ''-16.299''
|-
| [[33/10]]
| +9.695
| +16.340
|- style="background-color: #cccccc;"
| ''[[32/5]]''
| ''-9.705''
| ''-16.357''
|-
| [[27/14]]
| -9.712
| -16.369
|-
| [[21/17]]
| -9.828
| -16.563
|-
| [[21/1]]
| +9.854
| +16.609
|- style="background-color: #cccccc;"
| ''[[13/8]]''
| ''-9.866''
| ''-16.628''
|-
| [[27/1]]
| -9.899
| -16.684
|-
| [[3/2]]
| +10.041
| +16.923
|-
| [[28/3]]
| -10.227
| -17.237
|-
| [[17/13]]
| +10.236
| +17.252
|-
| [[22/15]]
| -10.386
| -17.505
|-
| [[15/13]]
| -10.409
| -17.544
|-
| [[33/31]]
| +10.429
| +17.576
|- style="background-color: #cccccc;"
| ''[[23/17]]''
| ''+10.678''
| ''+17.997''
|-
| [[33/13]]
| -10.755
| -18.126
|-
| [[31/15]]
| -10.774
| -18.159
|-
| [[7/5]]
| +10.818
| +18.232
|- style="background-color: #cccccc;"
| ''[[24/19]]''
| ''+10.889''
| ''+18.352''
|-
| [[10/1]]
| -11.004
| -18.546
|-
| [[23/8]]
| +11.048
| +18.620
|-
| [[29/26]]
| -11.051
| -18.625
|-
| [[11/7]]
| -11.163
| -18.815
|-
| [[25/9]]
| +11.272
| +18.998
|- style="background-color: #cccccc;"
| ''[[25/24]]''
| ''-11.339''
| ''-19.112''
|-
| [[22/1]]
| -11.350
| -19.129
|-
| [[31/14]]
| -11.551
| -19.468
|-
| [[29/3]]
| -11.645
| -19.627
|-
| [[19/9]]
| +11.723
| +19.757
|-
| [[29/4]]
| +11.736
| +19.779
|-
| '''[[31/1]]'''
| '''-11.738'''
| '''-19.782'''
|-
| [[27/11]]
| -11.890
| -20.039
|-
| [[33/2]]
| +12.031
| +20.278
|- style="background-color: #cccccc;"
| ''[[32/25]]''
| ''-12.042''
| ''-20.295''
|-
| [[33/28]]
| +12.218
| +20.592
|-
| [[27/5]]
| -12.235
| -20.621
|-
| [[23/6]]
| -12.333
| -20.786
|-
| [[15/2]]
| +12.377
| +20.860
|- style="background-color: #cccccc;"
| ''[[32/19]]''
| ''-12.492''
| ''-21.055''
|-
| [[28/15]]
| -12.564
| -21.175
|- style="background-color: #cccccc;"
| ''[[16/9]]''
| ''+12.571''
| ''+21.187''
|-
| [[31/20]]
| +12.607
| +21.248
|-
| [[13/3]]
| +12.746
| +21.481
|- style="background-color: #cccccc;"
| ''[[17/4]]''
| ''-12.970''
| ''-21.860''
|-
| [[11/10]]
| +12.995
| +21.901
|-
| '''[[7/1]]'''
| '''+13.154'''
| '''+22.170'''
|-
| '''[[2/1]]'''
| '''-13.340'''
| '''-22.484'''
|-
| [[28/1]]
| -13.527
| -22.798
|-
| [[33/29]]
| +13.636
| +22.982
|- style="background-color: #cccccc;"
| ''[[24/5]]''
| ''+13.676''
| ''+23.049''
|-
| [[22/5]]
| -13.686
| -23.067
|- style="background-color: #cccccc;"
| ''[[17/16]]''
| ''+13.711''
| ''+23.108''
|-
| [[31/11]]
| -13.728
| -23.138
|-
| [[26/21]]
| -13.749
| -23.172
|-
| [[29/15]]
| -13.982
| -23.565
|- style="background-color: #cccccc;"
| ''[[24/11]]''
| ''+14.021''
| ''+23.632''
|-
| [[29/23]]
| +14.028
| +23.643
|-
| [[31/5]]
| -14.074
| -23.720
|-
| [[15/7]]
| -14.117
| -23.793
|- style="background-color: #cccccc;"
| ''[[19/8]]''
| ''-14.188''
| ''-23.913''
|-
| [[30/1]]
| -14.304
| -24.107
|-
| [[24/23]]
| -14.348
| -24.182
|-
| [[27/20]]
| +14.446
| +24.347
|-
| [[33/7]]
| -14.463
| -24.376
|-
| [[19/17]]
| -14.559
| -24.537
|-
| [[27/25]]
| -14.572
| -24.559
|- style="background-color: #cccccc;"
| ''[[25/8]]''
| ''-14.639''
| ''-24.673''
|-
| [[30/23]]
| +14.669
| +24.724
|-
| [[29/14]]
| -14.759
| -24.874
|-
| [[31/4]]
| +14.943
| +25.185
|-
| '''[[29/1]]'''
| '''-14.945'''
| '''-25.189'''
|-
| [[25/17]]
| -15.009
| -25.297
|-
| [[27/19]]
| -15.022
| -25.318
|-
| [[29/12]]
| +15.035
| +25.341
|- style="background-color: #cccccc;"
| ''[[20/17]]''
| ''+15.307''
| ''+25.798''
|-
| [[11/2]]
| +15.331
| +25.839
|-
| [[28/23]]
| +15.446
| +26.033
|-
| [[28/11]]
| -15.517
| -26.153
|-
| [[23/2]]
| -15.633
| -26.347
|-
| [[5/2]]
| +15.677
| +26.422
|-
| [[28/5]]
| -15.863
| -26.736
|- style="background-color: #cccccc;"
| ''[[27/16]]''
| ''-15.870''
| ''-26.748''
|- style="background-color: #cccccc;"
| ''[[24/1]]''
| ''+16.012''
| ''+26.987''
|-
| [[25/22]]
| +16.022
| +27.004
|-
| [[13/9]]
| +16.045
| +27.043
|-
| [[19/10]]
| +16.127
| +27.181
|- style="background-color: #cccccc;"
| ''[[12/7]]''
| ''+16.199''
| ''+27.301''
|-
| [[30/11]]
| -16.294
| -27.463
|-
| [[31/25]]
| -16.410
| -27.658
|-
| [[7/3]]
| +16.454
| +27.731
|-
| [[22/19]]
| -16.473
| -27.764
|-
| [[6/1]]
| -16.640
| -28.045
|-
| [[27/4]]
| +16.782
| +28.284
|- style="background-color: #cccccc;"
| ''[[32/13]]''
| ''-16.815''
| ''-28.340''
|-
| [[31/19]]
| -16.861
| -28.417
|-
| [[29/11]]
| -16.936
| -28.544
|- style="background-color: #cccccc;"
| ''[[8/5]]''
| ''+16.975''
| ''+28.610''
|-
| [[26/7]]
| -17.048
| -28.734
|- style="background-color: #cccccc;"
| ''[[23/7]]''
| ''+17.206''
| ''+28.999''
|- style="background-color: #cccccc;"
| ''[[32/21]]''
| ''-17.223''
| ''-29.028''
|-
| [[31/23]]
| +17.236
| +29.049
|-
| [[29/5]]
| -17.281
| -29.126
|- style="background-color: #cccccc;"
| ''[[11/8]]''
| ''-17.321''
| ''-29.193''
|-
| [[17/5]]
| +17.346
| +29.234
|-
| [[23/22]]
| -17.623
| -29.703
|-
| [[17/11]]
| +17.691
| +29.817
|- style="background-color: #cccccc;"
| ''[[31/16]]''
| ''-17.709''
| ''-29.847''
|-
| [[20/9]]
| -17.745
| -29.908
|-
| [[23/10]]
| -17.969
| -30.285
|-
| [[25/2]]
| +18.013
| +30.359
|-
| [[28/25]]
| -18.200
| -30.674
|-
| [[31/12]]
| +18.243
| +30.747
|-
| [[19/2]]
| +18.464
| +31.119
|-
| [[11/6]]
| +18.631
| +31.400
|-
| [[28/19]]
| -18.650
| -31.433
|-
| [[6/5]]
| -18.976
| -31.983
|-
| [[27/23]]
| +19.074
| +32.148
|- style="background-color: #cccccc;"
| ''[[8/1]]''
| ''+19.312''
| ''+32.548''
|-
| [[27/13]]
| -19.345
| -32.604
|-
| [[30/19]]
| -19.427
| -32.742
|- style="background-color: #cccccc;"
| ''[[7/4]]''
| ''-19.498''
| ''-32.862''
|-
| [[29/25]]
| -19.618
| -33.064
|-
| '''[[17/1]]'''
| '''+19.682'''
| '''+33.172'''
|- style="background-color: #cccccc;"
| ''[[18/17]]''
| ''+19.711''
| ''+33.222''
|-
| [[9/7]]
| -19.753
| -33.292
|-
| [[17/14]]
| +19.868
| +33.486
|- style="background-color: #cccccc;"
| ''[[13/12]]''
| ''-19.907''
| ''-33.551''
|-
| [[18/1]]
| -19.940
| -33.606
|-
| [[29/19]]
| -20.068
| -33.823
|-
| [[9/4]]
| +20.082
| +33.845
|- style="background-color: #cccccc;"
| ''[[15/8]]''
| ''-20.275''
| ''-34.172''
|-
| [[13/10]]
| +20.450
| +34.466
|- style="background-color: #cccccc;"
| ''[[23/21]]''
| ''+20.506''
| ''+34.560''
|- style="background-color: #cccccc;"
| ''[[32/7]]''
| ''-20.523''
| ''-34.589''
|- style="background-color: #cccccc;"
| ''[[33/8]]''
| ''-20.621''
| ''-34.754''
|-
| [[17/15]]
| +20.645
| +34.796
|-
| [[22/13]]
| -20.796
| -35.049
|-
| [[21/10]]
| +20.858
| +35.155
|- style="background-color: #cccccc;"
| ''[[23/13]]''
| ''+20.914''
| ''+35.249''
|- style="background-color: #cccccc;"
| ''[[29/16]]''
| ''-20.917''
| ''-35.253''
|-
| [[33/17]]
| -20.991
| -35.378
|-
| [[20/3]]
| -21.045
| -35.469
|-
| [[31/13]]
| -21.183
| -35.703
|-
| [[22/21]]
| -21.204
| -35.737
|-
| [[25/6]]
| +21.313
| +35.921
|-
| [[31/21]]
| -21.592
| -36.391
|- style="background-color: #cccccc;"
| ''[[32/23]]''
| ''+21.604''
| ''+36.412''
|-
| [[19/6]]
| +21.763
| +36.680
|- style="background-color: #cccccc;"
| ''[[20/7]]''
| ''+21.835''
| ''+36.800''
|-
| [[18/11]]
| -21.930
| -36.961
|-
| [[18/5]]
| -22.276
| -37.544
|-
| [[23/9]]
| -22.374
| -37.709
|- style="background-color: #cccccc;"
| ''[[8/3]]''
| ''+22.611''
| ''+38.109''
|-
| [[13/2]]
| +22.786
| +38.404
|- style="background-color: #cccccc;"
| ''[[21/4]]''
| ''-22.798''
| ''-38.424''
|-
| [[28/13]]
| -22.973
| -38.718
|-
| [[17/3]]
| +22.982
| +38.733
|- style="background-color: #cccccc;"
| ''[[17/6]]''
| ''-23.011''
| ''-38.783''
|-
| [[33/20]]
| +23.035
| +38.824
|-
| [[27/7]]
| -23.053
| -38.853
|-
| [[21/2]]
| +23.195
| +39.093
|- style="background-color: #cccccc;"
| ''[[13/4]]''
| ''-23.206''
| ''-39.112''
|-
| [[4/3]]
| -23.381
| -39.407
|-
| [[26/17]]
| -23.576
| -39.736
|-
| [[30/13]]
| -23.750
| -40.028
|-
| [[10/7]]
| -24.158
| -40.716
|- style="background-color: #cccccc;"
| ''[[19/12]]''
| ''-24.229''
| ''-40.836''
|-
| [[20/1]]
| -24.344
| -41.030
|-
| [[23/16]]
| +24.388
| +41.104
|-
| [[29/13]]
| -24.391
| -41.109
|-
| [[22/7]]
| -24.504
| -41.299
|-
| [[25/18]]
| +24.612
| +41.482
|- style="background-color: #cccccc;"
| ''[[25/12]]''
| ''-24.680''
| ''-41.595''
|- style="background-color: #cccccc;"
| ''[[29/17]]''
| ''+24.706''
| ''+41.639''
|-
| [[29/21]]
| -24.799
| -41.797
|-
| [[31/7]]
| -24.891
| -41.952
|-
| [[19/18]]
| +25.063
| +42.241
|-
| [[29/8]]
| +25.076
| +42.263
|-
| [[26/23]]
| +25.079
| +42.268
|- style="background-color: #cccccc;"
| ''[[21/20]]''
| ''-25.134''
| ''-42.361''
|- style="background-color: #cccccc;"
| ''[[23/19]]''
| ''+25.237''
| ''+42.534''
|- style="background-color: #cccccc;"
| ''[[30/17]]''
| ''+25.347''
| ''+42.721''
|-
| [[33/4]]
| +25.372
| +42.762
|- style="background-color: #cccccc;"
| ''[[20/13]]''
| ''+25.543''
| ''+43.050''
|-
| [[23/3]]
| -25.673
| -43.270
|- style="background-color: #cccccc;"
| ''[[25/23]]''
| ''-25.687''
| ''-43.293''
|-
| [[15/4]]
| +25.718
| +43.344
|- style="background-color: #cccccc;"
| ''[[9/8]]''
| ''-25.911''
| ''-43.671''
|-
| [[13/6]]
| +26.086
| +43.965
|- style="background-color: #cccccc;"
| ''[[28/17]]''
| ''+26.124''
| ''+44.030''
|- style="background-color: #cccccc;"
| ''[[18/7]]''
| ''+26.239''
| ''+44.224''
|-
| [[17/9]]
| +26.281
| +44.294
|- style="background-color: #cccccc;"
| ''[[17/2]]''
| ''-26.311''
| ''-44.344''
|-
| [[20/11]]
| -26.335
| -44.385
|-
| [[7/2]]
| +26.494
| +44.654
|-
| [[4/1]]
| -26.681
| -44.968
|- style="background-color: #cccccc;"
| ''[[12/5]]''
| ''+27.016''
| ''+45.533''
|- style="background-color: #cccccc;"
| ''[[32/17]]''
| ''-27.051''
| ''-45.592''
|- style="background-color: #cccccc;"
| ''[[12/11]]''
| ''+27.362''
| ''+46.116''
|-
| [[30/7]]
| -27.458
| -46.277
|- style="background-color: #cccccc;"
| ''[[19/4]]''
| ''-27.529''
| ''-46.397''
|-
| [[33/23]]
| +27.664
| +46.625
|- style="background-color: #cccccc;"
| ''[[31/24]]''
| ''-27.750''
| ''-46.769''
|- style="background-color: #cccccc;"
| ''[[31/17]]''
| ''+27.913''
| ''+47.045''
|- style="background-color: #cccccc;"
| ''[[25/4]]''
| ''-27.979''
| ''-47.157''
|-
| [[23/15]]
| -28.010
| -47.208
|- style="background-color: #cccccc;"
| ''[[23/5]]''
| ''+28.023''
| ''+47.231''
|-
| [[29/7]]
| -28.099
| -47.358
|-
| [[31/8]]
| +28.284
| +47.669
|- style="background-color: #cccccc;"
| ''[[22/17]]''
| ''+28.301''
| ''+47.699''
|- style="background-color: #cccccc;"
| ''[[23/11]]''
| ''+28.369''
| ''+47.813''
|-
| [[29/24]]
| +28.376
| +47.824
|- style="background-color: #cccccc;"
| ''[[17/10]]''
| ''-28.647''
| ''-48.282''
|-
| [[11/4]]
| +28.671
| +48.323
|-
| [[23/14]]
| -28.787
| -48.517
|-
| '''[[23/1]]'''
| '''-28.973'''
| '''-48.831'''
|-
| [[5/4]]
| +29.017
| +48.906
|- style="background-color: #cccccc;"
| ''[[27/8]]''
| ''-29.211''
| ''-49.232''
|- style="background-color: #cccccc;"
| ''[[12/1]]''
| ''+29.353''
| ''+49.471''
|-
| [[18/13]]
| -29.386
| -49.526
|-
| [[20/19]]
| -29.468
| -49.665
|- style="background-color: #cccccc;"
| ''[[7/6]]''
| ''-29.539''
| ''-49.785''
|-
| [[27/17]]
| -29.581
| -49.856
|}
 
{| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 33-integer-limit intervals in 71zpi (by patent val mapping)
|-
! Ratio
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
| [[14/1]]
| -0.186
| -0.314
|-
| [[11/5]]
| -0.346
| -0.583
|-
| [[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
|-
| [[15/14]]
| -0.777
| -1.309
|-
| [[15/1]]
| -0.963
| -1.623
|-
| [[23/12]]
| +1.007
| +1.698
|-
| [[27/10]]
| +1.105
| +1.863
|-
| [[33/14]]
| -1.123
| -1.892
|-
| [[33/1]]
| -1.309
| -2.206
|-
| [[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
|-
| [[31/28]]
| +1.789
| +3.016
|-
| [[31/27]]
| -1.839
| -3.099
|-
| '''[[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
|-
| [[31/30]]
| +2.566
| +4.325
|-
| [[33/26]]
| +2.586
| +4.358
|-
| [[25/11]]
| +2.682
| +4.520
|-
| [[26/9]]
| +2.705
| +4.559
|-
| [[19/5]]
| +2.787
| +4.697
|-
| [[26/15]]
| -2.931
| -4.940
|-
| [[15/11]]
| -2.954
| -4.979
|-
| [[14/3]]
| +3.113
| +5.247
|-
| [[19/11]]
| +3.133
| +5.280
|-
| [[31/29]]
| +3.208
| +5.406
|-
| '''[[3/1]]'''
| '''-3.300'''
| '''-5.561'''
|-
| [[27/2]]
| +3.442
| +5.800
|-
| [[29/22]]
| -3.595
| -6.060
|-
| [[28/27]]
| -3.628
| -6.115
|-
| [[33/5]]
| -3.645
| -6.144
|-
| [[13/7]]
| -3.708
| -6.250
|-
| [[26/1]]
| -3.894
| -6.564
|-
| [[29/10]]
| -3.941
| -6.642
|-
| [[19/13]]
| -4.323
| -7.285
|-
| [[10/9]]
| -4.405
| -7.424
|-
| [[23/20]]
| -4.629
| -7.801
|-
| [[25/1]]
| +4.673
| +7.875
|-
| [[21/19]]
| +4.731
| +7.974
|-
| [[22/9]]
| -4.750
| -8.006
|-
| [[25/13]]
| -4.773
| -8.045
|-
| [[25/14]]
| +4.859
| +8.190
|-
| [[31/6]]
| +4.903
| +8.263
|-
| [[29/18]]
| +4.995
| +8.418
|-
| [[29/27]]
| -5.046
| -8.505
|-
| '''[[19/1]]'''
| '''+5.123'''
| '''+8.635'''
|-
| [[31/9]]
| -5.138
| -8.660
|-
| [[25/21]]
| -5.182
| -8.733
|-
| [[11/3]]
| +5.290
| +8.916
|-
| [[19/14]]
| +5.310
| +8.949
|-
| [[5/3]]
| +5.636
| +9.499
|-
| [[26/11]]
| -5.885
| -9.919
|-
| [[33/25]]
| -5.982
| -10.082
|-
| [[27/26]]
| -6.004
| -10.120
|-
| [[19/15]]
| +6.087
| +10.258
|-
| [[26/5]]
| -6.231
| -10.502
|-
| [[14/9]]
| +6.413
| +10.808
|-
| [[33/19]]
| -6.432
| -10.841
|-
| [[17/7]]
| +6.528
| +11.002
|-
| [[9/1]]
| -6.599
| -11.122
|-
| [[9/2]]
| +6.741
| +11.362
|-
| [[28/9]]
| -6.928
| -11.676
|-
| [[13/5]]
| +7.110
| +11.982
|-
| [[13/11]]
| +7.455
| +12.565
|-
| [[21/5]]
| +7.518
| +12.671
|-
| [[10/3]]
| -7.704
| -12.985
|-
| [[31/26]]
| -7.843
| -13.219
|-
| [[21/11]]
| +7.864
| +13.253
|-
| [[25/3]]
| +7.972
| +13.437
|-
| [[19/7]]
| -8.031
| -13.535
|-
| [[22/3]]
| -8.050
| -13.568
|-
| [[31/18]]
| +8.202
| +13.824
|-
| [[29/9]]
| -8.346
| -14.066
|-
| [[19/3]]
| +8.423
| +14.196
|-
| [[31/3]]
| -8.438
| -14.221
|-
| [[25/7]]
| -8.481
| -14.294
|-
| [[26/25]]
| -8.567
| -14.439
|-
| [[11/9]]
| +8.590
| +14.478
|-
| [[9/5]]
| -8.936
| -15.060
|-
| [[26/19]]
| -9.018
| -15.199
|-
| [[23/18]]
| -9.033
| -15.225
|-
| [[29/20]]
| +9.399
| +15.842
|-
| '''[[13/1]]'''
| '''+9.446'''
| '''+15.920'''
|-
| [[14/13]]
| -9.632
| -16.234
|-
| [[33/10]]
| +9.695
| +16.340
|-
| [[27/14]]
| -9.712
| -16.369
|-
| [[21/17]]
| -9.828
| -16.563
|-
| [[21/1]]
| +9.854
| +16.609
|-
| [[27/1]]
| -9.899
| -16.684
|-
| [[3/2]]
| +10.041
| +16.923
|-
| [[28/3]]
| -10.227
| -17.237
|-
| [[17/13]]
| +10.236
| +17.252
|-
| [[22/15]]
| -10.386
| -17.505
|-
| [[15/13]]
| -10.409
| -17.544
|-
| [[33/31]]
| +10.429
| +17.576
|-
| [[33/13]]
| -10.755
| -18.126
|-
| [[31/15]]
| -10.774
| -18.159
|-
| [[7/5]]
| +10.818
| +18.232
|-
| [[10/1]]
| -11.004
| -18.546
|-
| [[23/8]]
| +11.048
| +18.620
|-
| [[29/26]]
| -11.051
| -18.625
|-
| [[11/7]]
| -11.163
| -18.815
|-
| [[25/9]]
| +11.272
| +18.998
|-
| [[22/1]]
| -11.350
| -19.129
|-
| [[31/14]]
| -11.551
| -19.468
|-
| [[29/3]]
| -11.645
| -19.627
|-
| [[19/9]]
| +11.723
| +19.757
|-
| [[29/4]]
| +11.736
| +19.779
|-
| '''[[31/1]]'''
| '''-11.738'''
| '''-19.782'''
|-
| [[27/11]]
| -11.890
| -20.039
|-
| [[33/2]]
| +12.031
| +20.278
|-
| [[33/28]]
| +12.218
| +20.592
|-
| [[27/5]]
| -12.235
| -20.621
|-
| [[23/6]]
| -12.333
| -20.786
|-
| [[15/2]]
| +12.377
| +20.860
|-
| [[28/15]]
| -12.564
| -21.175
|-
| [[31/20]]
| +12.607
| +21.248
|-
| [[13/3]]
| +12.746
| +21.481
|-
| [[11/10]]
| +12.995
| +21.901
|-
| '''[[7/1]]'''
| '''+13.154'''
| '''+22.170'''
|-
| '''[[2/1]]'''
| '''-13.340'''
| '''-22.484'''
|-
| [[28/1]]
| -13.527
| -22.798
|-
| [[33/29]]
| +13.636
| +22.982
|-
| [[22/5]]
| -13.686
| -23.067
|-
| [[31/11]]
| -13.728
| -23.138
|-
| [[26/21]]
| -13.749
| -23.172
|-
| [[29/15]]
| -13.982
| -23.565
|-
| [[29/23]]
| +14.028
| +23.643
|-
| [[31/5]]
| -14.074
| -23.720
|-
| [[15/7]]
| -14.117
| -23.793
|-
| [[30/1]]
| -14.304
| -24.107
|-
| [[24/23]]
| -14.348
| -24.182
|-
| [[27/20]]
| +14.446
| +24.347
|-
| [[33/7]]
| -14.463
| -24.376
|-
| [[19/17]]
| -14.559
| -24.537
|-
| [[27/25]]
| -14.572
| -24.559
|-
| [[30/23]]
| +14.669
| +24.724
|-
| [[29/14]]
| -14.759
| -24.874
|-
| [[31/4]]
| +14.943
| +25.185
|-
| '''[[29/1]]'''
| '''-14.945'''
| '''-25.189'''
|-
| [[25/17]]
| -15.009
| -25.297
|-
| [[27/19]]
| -15.022
| -25.318
|-
| [[29/12]]
| +15.035
| +25.341
|-
| [[11/2]]
| +15.331
| +25.839
|-
| [[28/23]]
| +15.446
| +26.033
|-
| [[28/11]]
| -15.517
| -26.153
|-
| [[23/2]]
| -15.633
| -26.347
|-
| [[5/2]]
| +15.677
| +26.422
|-
| [[28/5]]
| -15.863
| -26.736
|-
| [[25/22]]
| +16.022
| +27.004
|-
| [[13/9]]
| +16.045
| +27.043
|-
| [[19/10]]
| +16.127
| +27.181
|-
| [[30/11]]
| -16.294
| -27.463
|-
| [[31/25]]
| -16.410
| -27.658
|-
| [[7/3]]
| +16.454
| +27.731
|-
| [[22/19]]
| -16.473
| -27.764
|-
| [[6/1]]
| -16.640
| -28.045
|-
| [[27/4]]
| +16.782
| +28.284
|-
| [[31/19]]
| -16.861
| -28.417
|-
| [[29/11]]
| -16.936
| -28.544
|-
| [[26/7]]
| -17.048
| -28.734
|-
| [[31/23]]
| +17.236
| +29.049
|-
| [[29/5]]
| -17.281
| -29.126
|-
| [[17/5]]
| +17.346
| +29.234
|-
| [[23/22]]
| -17.623
| -29.703
|-
| [[17/11]]
| +17.691
| +29.817
|-
| [[20/9]]
| -17.745
| -29.908
|-
| [[23/10]]
| -17.969
| -30.285
|-
| [[25/2]]
| +18.013
| +30.359
|-
| [[28/25]]
| -18.200
| -30.674
|-
| [[31/12]]
| +18.243
| +30.747
|-
| [[19/2]]
| +18.464
| +31.119
|-
| [[11/6]]
| +18.631
| +31.400
|-
| [[28/19]]
| -18.650
| -31.433
|-
| [[6/5]]
| -18.976
| -31.983
|-
| [[27/23]]
| +19.074
| +32.148
|-
| [[27/13]]
| -19.345
| -32.604
|-
| [[30/19]]
| -19.427
| -32.742
|-
| [[29/25]]
| -19.618
| -33.064
|-
| '''[[17/1]]'''
| '''+19.682'''
| '''+33.172'''
|-
| [[9/7]]
| -19.753
| -33.292
|-
| [[17/14]]
| +19.868
| +33.486
|-
| [[18/1]]
| -19.940
| -33.606
|-
| [[29/19]]
| -20.068
| -33.823
|-
| [[9/4]]
| +20.082
| +33.845
|-
| [[13/10]]
| +20.450
| +34.466
|-
| [[17/15]]
| +20.645
| +34.796
|-
| [[22/13]]
| -20.796
| -35.049
|-
| [[21/10]]
| +20.858
| +35.155
|-
| [[33/17]]
| -20.991
| -35.378
|-
| [[20/3]]
| -21.045
| -35.469
|-
| [[31/13]]
| -21.183
| -35.703
|-
| [[22/21]]
| -21.204
| -35.737
|-
| [[25/6]]
| +21.313
| +35.921
|-
| [[31/21]]
| -21.592
| -36.391
|-
| [[19/6]]
| +21.763
| +36.680
|-
| [[18/11]]
| -21.930
| -36.961
|-
| [[18/5]]
| -22.276
| -37.544
|-
| [[23/9]]
| -22.374
| -37.709
|-
| [[13/2]]
| +22.786
| +38.404
|-
| [[28/13]]
| -22.973
| -38.718
|-
| [[17/3]]
| +22.982
| +38.733
|-
| [[33/20]]
| +23.035
| +38.824
|-
| [[27/7]]
| -23.053
| -38.853
|-
| [[21/2]]
| +23.195
| +39.093
|-
| [[4/3]]
| -23.381
| -39.407
|-
| [[26/17]]
| -23.576
| -39.736
|-
| [[30/13]]
| -23.750
| -40.028
|-
| [[10/7]]
| -24.158
| -40.716
|-
| [[20/1]]
| -24.344
| -41.030
|-
| [[23/16]]
| +24.388
| +41.104
|-
| [[29/13]]
| -24.391
| -41.109
|-
| [[22/7]]
| -24.504
| -41.299
|-
| [[25/18]]
| +24.612
| +41.482
|-
| [[29/21]]
| -24.799
| -41.797
|-
| [[31/7]]
| -24.891
| -41.952
|-
| [[19/18]]
| +25.063
| +42.241
|-
| [[29/8]]
| +25.076
| +42.263
|-
| [[26/23]]
| +25.079
| +42.268
|-
| [[33/4]]
| +25.372
| +42.762
|-
| [[23/3]]
| -25.673
| -43.270
|-
| [[15/4]]
| +25.718
| +43.344
|-
| [[13/6]]
| +26.086
| +43.965
|-
| [[17/9]]
| +26.281
| +44.294
|-
| [[20/11]]
| -26.335
| -44.385
|-
| [[7/2]]
| +26.494
| +44.654
|-
| [[4/1]]
| -26.681
| -44.968
|-
| [[30/7]]
| -27.458
| -46.277
|-
| [[33/23]]
| +27.664
| +46.625
|-
| [[23/15]]
| -28.010
| -47.208
|-
| [[29/7]]
| -28.099
| -47.358
|-
| [[31/8]]
| +28.284
| +47.669
|-
| [[29/24]]
| +28.376
| +47.824
|-
| [[11/4]]
| +28.671
| +48.323
|-
| [[23/14]]
| -28.787
| -48.517
|-
| '''[[23/1]]'''
| '''-28.973'''
| '''-48.831'''
|-
| [[5/4]]
| +29.017
| +48.906
|-
| [[18/13]]
| -29.386
| -49.526
|-
| [[20/19]]
| -29.468
| -49.665
|-
| [[27/17]]
| -29.581
| -49.856
|- style="background-color: #cccccc;"
| ''[[7/6]]''
| ''+29.794''
| ''+50.215''
|- style="background-color: #cccccc;"
| ''[[12/1]]''
| ''-29.980''
| ''-50.529''
|- style="background-color: #cccccc;"
| ''[[27/8]]''
| ''+30.122''
| ''+50.768''
|- style="background-color: #cccccc;"
| ''[[17/10]]''
| ''+30.686''
| ''+51.718''
|- style="background-color: #cccccc;"
| ''[[23/11]]''
| ''-30.964''
| ''-52.187''
|- style="background-color: #cccccc;"
| ''[[22/17]]''
| ''-31.032''
| ''-52.301''
|- style="background-color: #cccccc;"
| ''[[23/5]]''
| ''-31.309''
| ''-52.769''
|- style="background-color: #cccccc;"
| ''[[25/4]]''
| ''+31.354''
| ''+52.843''
|- style="background-color: #cccccc;"
| ''[[31/17]]''
| ''-31.419''
| ''-52.955''
|- style="background-color: #cccccc;"
| ''[[31/24]]''
| ''+31.583''
| ''+53.231''
|- style="background-color: #cccccc;"
| ''[[19/4]]''
| ''+31.804''
| ''+53.603''
|- style="background-color: #cccccc;"
| ''[[12/11]]''
| ''-31.971''
| ''-53.884''
|- style="background-color: #cccccc;"
| ''[[12/5]]''
| ''-32.317''
| ''-54.467''
|- style="background-color: #cccccc;"
| ''[[17/2]]''
| ''+33.022''
| ''+55.656''
|- style="background-color: #cccccc;"
| ''[[18/7]]''
| ''-33.094''
| ''-55.776''
|- style="background-color: #cccccc;"
| ''[[28/17]]''
| ''-33.209''
| ''-55.970''
|- style="background-color: #cccccc;"
| ''[[9/8]]''
| ''+33.422''
| ''+56.329''
|- style="background-color: #cccccc;"
| ''[[25/23]]''
| ''+33.646''
| ''+56.707''
|- style="background-color: #cccccc;"
| ''[[20/13]]''
| ''-33.790''
| ''-56.950''
|- style="background-color: #cccccc;"
| ''[[30/17]]''
| ''-33.986''
| ''-57.279''
|- style="background-color: #cccccc;"
| ''[[23/19]]''
| ''-34.096''
| ''-57.466''
|- style="background-color: #cccccc;"
| ''[[21/20]]''
| ''+34.199''
| ''+57.639''
|- style="background-color: #cccccc;"
| ''[[29/17]]''
| ''-34.627''
| ''-58.361''
|- style="background-color: #cccccc;"
| ''[[25/12]]''
| ''+34.653''
| ''+58.405''
|- style="background-color: #cccccc;"
| ''[[19/12]]''
| ''+35.104''
| ''+59.164''
|- style="background-color: #cccccc;"
| ''[[13/4]]''
| ''+36.127''
| ''+60.888''
|- style="background-color: #cccccc;"
| ''[[17/6]]''
| ''+36.322''
| ''+61.217''
|- style="background-color: #cccccc;"
| ''[[21/4]]''
| ''+36.535''
| ''+61.576''
|- style="background-color: #cccccc;"
| ''[[8/3]]''
| ''-36.722''
| ''-61.891''
|- style="background-color: #cccccc;"
| ''[[20/7]]''
| ''-37.498''
| ''-63.200''
|- style="background-color: #cccccc;"
| ''[[32/23]]''
| ''-37.729''
| ''-63.588''
|- style="background-color: #cccccc;"
| ''[[29/16]]''
| ''+38.416''
| ''+64.747''
|- style="background-color: #cccccc;"
| ''[[23/13]]''
| ''-38.419''
| ''-64.751''
|- style="background-color: #cccccc;"
| ''[[33/8]]''
| ''+38.712''
| ''+65.246''
|- style="background-color: #cccccc;"
| ''[[23/21]]''
| ''-38.827''
| ''-65.440''
|- style="background-color: #cccccc;"
| ''[[15/8]]''
| ''+39.058''
| ''+65.828''
|- style="background-color: #cccccc;"
| ''[[13/12]]''
| ''+39.426''
| ''+66.449''
|- style="background-color: #cccccc;"
| ''[[18/17]]''
| ''-39.622''
| ''-66.778''
|- style="background-color: #cccccc;"
| ''[[7/4]]''
| ''+39.835''
| ''+67.138''
|- style="background-color: #cccccc;"
| ''[[8/1]]''
| ''-40.021''
| ''-67.452''
|- style="background-color: #cccccc;"
| ''[[31/16]]''
| ''+41.624''
| ''+70.153''
|- style="background-color: #cccccc;"
| ''[[11/8]]''
| ''+42.012''
| ''+70.807''
|- style="background-color: #cccccc;"
| ''[[23/7]]''
| ''-42.127''
| ''-71.001''
|- style="background-color: #cccccc;"
| ''[[8/5]]''
| ''-42.358''
| ''-71.390''
|- style="background-color: #cccccc;"
| ''[[12/7]]''
| ''-43.134''
| ''-72.699''
|- style="background-color: #cccccc;"
| ''[[24/1]]''
| ''-43.321''
| ''-73.013''
|- style="background-color: #cccccc;"
| ''[[27/16]]''
| ''+43.463''
| ''+73.252''
|- style="background-color: #cccccc;"
| ''[[20/17]]''
| ''-44.026''
| ''-74.202''
|- style="background-color: #cccccc;"
| ''[[25/8]]''
| ''+44.694''
| ''+75.327''
|- style="background-color: #cccccc;"
| ''[[19/8]]''
| ''+45.144''
| ''+76.087''
|- style="background-color: #cccccc;"
| ''[[24/11]]''
| ''-45.311''
| ''-76.368''
|- style="background-color: #cccccc;"
| ''[[24/5]]''
| ''-45.657''
| ''-76.951''
|- style="background-color: #cccccc;"
| ''[[17/4]]''
| ''+46.363''
| ''+78.140''
|- style="background-color: #cccccc;"
| ''[[16/9]]''
| ''-46.762''
| ''-78.813''
|- style="background-color: #cccccc;"
| ''[[25/24]]''
| ''+47.994''
| ''+80.888''
|- style="background-color: #cccccc;"
| ''[[24/19]]''
| ''-48.444''
| ''-81.648''
|- style="background-color: #cccccc;"
| ''[[23/17]]''
| ''-48.655''
| ''-82.003''
|- style="background-color: #cccccc;"
| ''[[13/8]]''
| ''+49.467''
| ''+83.372''
|- style="background-color: #cccccc;"
| ''[[17/12]]''
| ''+49.662''
| ''+83.701''
|- style="background-color: #cccccc;"
| ''[[21/8]]''
| ''+49.876''
| ''+84.060''
|- style="background-color: #cccccc;"
| ''[[16/3]]''
| ''-50.062''
| ''-84.375''
|- style="background-color: #cccccc;"
| ''[[32/29]]''
| ''-51.757''
| ''-87.231''
|- style="background-color: #cccccc;"
| ''[[33/16]]''
| ''+52.053''
| ''+87.730''
|- style="background-color: #cccccc;"
| ''[[16/15]]''
| ''-52.398''
| ''-88.312''
|- style="background-color: #cccccc;"
| ''[[24/13]]''
| ''-52.767''
| ''-88.933''
|- style="background-color: #cccccc;"
| ''[[8/7]]''
| ''-53.175''
| ''-89.622''
|- style="background-color: #cccccc;"
| ''[[16/1]]''
| ''-53.362''
| ''-89.936''
|- style="background-color: #cccccc;"
| ''[[32/31]]''
| ''-54.964''
| ''-92.637''
|- style="background-color: #cccccc;"
| ''[[16/11]]''
| ''-55.352''
| ''-93.291''
|- style="background-color: #cccccc;"
| ''[[16/5]]''
| ''-55.698''
| ''-93.873''
|- style="background-color: #cccccc;"
| ''[[24/7]]''
| ''-56.475''
| ''-95.183''
|- style="background-color: #cccccc;"
| ''[[32/27]]''
| ''-56.803''
| ''-95.736''
|- style="background-color: #cccccc;"
| ''[[25/16]]''
| ''+58.034''
| ''+97.811''
|- style="background-color: #cccccc;"
| ''[[19/16]]''
| ''+58.485''
| ''+98.571''
|- style="background-color: #cccccc;"
| ''[[17/8]]''
| ''+59.703''
| ''+100.624''
|- style="background-color: #cccccc;"
| ''[[32/9]]''
| ''-60.103''
| ''-101.297''
|- style="background-color: #cccccc;"
| ''[[16/13]]''
| ''-62.807''
| ''-105.856''
|- style="background-color: #cccccc;"
| ''[[24/17]]''
| ''-63.003''
| ''-106.185''
|- style="background-color: #cccccc;"
| ''[[21/16]]''
| ''+63.216''
| ''+106.544''
|- style="background-color: #cccccc;"
| ''[[32/3]]''
| ''-63.402''
| ''-106.858''
|- style="background-color: #cccccc;"
| ''[[33/32]]''
| ''+65.393''
| ''+110.214''
|- style="background-color: #cccccc;"
| ''[[32/15]]''
| ''-65.739''
| ''-110.796''
|- style="background-color: #cccccc;"
| ''[[16/7]]''
| ''-66.516''
| ''-112.106''
|- style="background-color: #cccccc;"
| ''[[32/1]]''
| ''-66.702''
| ''-112.420''
|- style="background-color: #cccccc;"
| ''[[32/11]]''
| ''-68.693''
| ''-115.775''
|- style="background-color: #cccccc;"
| ''[[32/5]]''
| ''-69.038''
| ''-116.357''
|- style="background-color: #cccccc;"
| ''[[32/25]]''
| ''-71.375''
| ''-120.295''
|- style="background-color: #cccccc;"
| ''[[32/19]]''
| ''-71.825''
| ''-121.055''
|- style="background-color: #cccccc;"
| ''[[17/16]]''
| ''+73.044''
| ''+123.108''
|- style="background-color: #cccccc;"
| ''[[32/13]]''
| ''-76.148''
| ''-128.340''
|- style="background-color: #cccccc;"
| ''[[32/21]]''
| ''-76.556''
| ''-129.028''
|- style="background-color: #cccccc;"
| ''[[32/7]]''
| ''-79.856''
| ''-134.589''
|- style="background-color: #cccccc;"
| ''[[32/17]]''
| ''-86.384''
| ''-145.592''
|}
 
== Record on the Riemann zeta function with prime 2 removed ==
'''[[71zpi]]''' sets a height record on the Riemann zeta function with prime 2 removed. The previous record is [[53zpi]] and the next one is [[93zpi]]. It is important to highlight that the optimal equal tunings obtained by excluding the prime number 2 from the Riemann zeta function differs slightly from the optimal equal tuning corresponding to the same peaks on the unmodified Riemann zeta function.
{| class="wikitable"
! colspan="6" |Unmodified Riemann zeta function
! colspan="5" |Riemann zeta function with prime 2 removed
|-
! colspan="3" | Tuning
! colspan="1" |Strength
! colspan="2" |Closest EDO
! colspan="2" |Tuning
! colspan="1" |Strength
! colspan="2" |Closest EDO
|-
|-
!ZPI
!ZPI
!Steps per octave
!Steps per octave
!Step size (cents)
!Step size (cents)
!Height
! colspan="1" | Height
!Integral
!Gap
!EDO
!EDO
!Octave (cents)
!Octave (cents)
!Consistent
!Steps per octave
!Distinct
!Step size (cents)
! colspan="1" |Height
!EDO
!Octave (cents)
|-
|[[53zpi]]
| 16.3979501311478
|73.1798786069366
|2.518818
| [[16edo]]
|1170.87805771099
| 16.4044889390925
|73.1507092025500
|4.100909
|[[16edo]]
|1170.41134724080
|-
|-
|[[71zpi]]
|[[71zpi]]
|20.2248393119540
|20.2248393119540
|59.3329806724710
|59.3329806724710
|3.531097
| 3.531097
|0.613581
|12.986080
|[[20edo]]
|[[20edo]]
|1186.65961344942
|1186.65961344942
|6
|20.2459529213541
|6
|59.2711049295348
|4.137236
|[[20edo]]
|1185.42209859070
|-
| [[93zpi]]
| 24.5782550666850
|48.8236449961234
|2.810487
|[[25edo]]
|1220.59112490308
|24.5738316304204
|48.8324335434323
|4.665720
|[[25edo]]
|1220.81083858581
|}
|}


[[File:71zpi.png|thumb|The Riemann zeta function around 71zpi]]
=== Harmonic series in 71zpi with prime 2 removed ===


== Theory ==
{{Harmonics in cet|59.2711049295348|columns=15|title=Approximation of harmonics in 71zpi with prime 2 removed }}
'''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.
{{Harmonics in cet|59.2711049295348|columns=18|start=16|title=Approximation of harmonics in 71zpi with prime 2 removed }}


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]].
=== Intervals in 71zpi with prime 2 removed ===


The nearest zeta peaks to 71zpi that surpass its strength are [[65zpi]] and [[75zpi]].
{| class="wikitable center-1 right-2 left-3 center-4 center-5"
|+ style="white-space:nowrap" | Intervals in 71zpi with prime 2 removed
|-
| colspan="3" style="text-align:left;" | JI ratios are comprised of 34-integer-limit ratios,<br>and are stylized as follows to indicate their accuracy:
* '''<u>Bold Underlined:</u>''' relative error < 8.333 %
* '''Bold:''' relative error < 16.667 %
* Normal: relative error < 25 %
* <small>Small:</small> relative error < 33.333 %
* <small><small>Small Small:</small></small> relative error < 41.667 %
* <small><small><small>Small Small Small:</small></small></small> relative error < 50 %
| colspan="2" style="text-align:right;" | <center>'''⟨81 128] at every 4 steps'''</center><br>[[9/8|Whole tone]] = 13 steps<br>[[256/243|Limma]] = 8 steps<br>[[2187/2048|Apotome]] = 5 steps
|-
! Degree
! Cents
! Ratios
! Ups and downs notation
! Step
|-
| 0
| 0.000
|
| P1
| 0
|-
| 1
| 59.271
| '''[[34/33]]''', '''[[33/32]]''', '''<u>[[32/31]]'''</u>, '''<u>[[31/30]]'''</u>, '''<u>[[30/29]]'''</u>, '''<u>[[29/28]]'''</u>, '''<u>[[28/27]]'''</u>, '''[[27/26]]''', '''[[26/25]]''', [[25/24]], [[24/23]], <small>[[23/22]]</small>, <small><small>[[22/21]]</small></small>, <small><small><small>[[21/20]]</small></small></small>, <small><small><small>[[20/19]]</small></small></small>
| vA1, ^d2
| 4
|-
| 2
| 118.542
| <small><small><small>[[19/18]]</small></small></small>, <small>[[18/17]]</small>, [[17/16]], [[33/31]], '''[[16/15]]''', '''<u>[[31/29]]'''</u>, '''<u>[[15/14]]'''</u>, '''[[29/27]]''', '''[[14/13]]''', [[27/25]], <small><small>[[13/12]]</small></small>, <small><small><small>[[25/23]]</small></small></small>
| m2
| 8
|-
| 3
| 177.813
| <small><small><small>[[12/11]]</small></small></small>, <small><small>[[23/21]]</small></small>, <small>[[34/31]]</small>, [[11/10]], '''[[32/29]]''', '''<u>[[21/19]]'''</u>, '''<u>[[31/28]]'''</u>, '''<u>[[10/9]]'''</u>, [[29/26]], [[19/17]], <small>[[28/25]]</small>, <small><small><small>[[9/8]]</small></small></small>
| vM2
| 12
|-
| 4
| 237.084
| <small><small><small>[[26/23]]</small></small></small>, <small><small>[[17/15]]</small></small>, <small>[[25/22]]</small>, [[33/29]], '''[[8/7]]''', '''<u>[[31/27]]'''</u>, '''<u>[[23/20]]'''</u>, [[15/13]], <small>[[22/19]]</small>, <small><small>[[29/25]]</small></small>
| vvA2
| 16
|-
| 5
| 296.356
| <small><small><small>[[7/6]]</small></small></small>, <small><small>[[34/29]]</small></small>, <small>[[27/23]]</small>, <small>[[20/17]]</small>, [[33/28]], '''[[13/11]]''', '''<u>[[32/27]]'''</u>, '''<u>[[19/16]]'''</u>, '''[[25/21]]''', '''[[31/26]]''', <small>[[6/5]]</small>
| vm3
| 20
|-
| 6
| 355.627
| <small><small><small>[[29/24]]</small></small></small>, <small><small><small>[[23/19]]</small></small></small>, <small>[[17/14]]</small>, <small>[[28/23]]</small>, '''[[11/9]]''', '''<u>[[27/22]]'''</u>, '''<u>[[16/13]]'''</u>, [[21/17]], [[26/21]], <small>[[31/25]]</small>
| vvM3
| 24
|-
| 7
| 414.898
| <small><small><small>[[5/4]]</small></small></small>, <small>[[34/27]]</small>, [[29/23]], [[24/19]], '''[[19/15]]''', '''<u>[[33/26]]'''</u>, '''<u>[[14/11]]'''</u>, '''[[23/18]]''', [[32/25]], <small><small>[[9/7]]</small></small>, <small><small><small>[[31/24]]</small></small></small>
| ^^M3
| 28
|-
| 8
| 474.169
| <small><small><small>[[22/17]]</small></small></small>, <small><small>[[13/10]]</small></small>, [[30/23]], '''[[17/13]]''', '''<u>[[21/16]]'''</u>, '''<u>[[25/19]]'''</u>, '''<u>[[29/22]]'''</u>, '''[[33/25]]''', <small><small>[[4/3]]</small></small>
| vv4
| 32
|-
| 9
| 533.440
| <small>[[31/23]]</small>, [[27/20]], [[23/17]], '''<u>[[19/14]]'''</u>, '''<u>[[34/25]]'''</u>, '''<u>[[15/11]]'''</u>, '''[[26/19]]''', <small>[[11/8]]</small>, <small><small><small>[[29/21]]</small></small></small>
| ^^4
| 36
|-
| 10
| 592.711
| <small><small><small>[[18/13]]</small></small></small>, <small><small>[[25/18]]</small></small>, <small><small>[[32/23]]</small></small>, [[7/5]], '''<u>[[31/22]]'''</u>, '''<u>[[24/17]]'''</u>, [[17/12]], <small>[[27/19]]</small>, <small><small><small>[[10/7]]</small></small></small>
| ^A4
| 40
|-
| 11
| 651.982
| <small><small><small>[[33/23]]</small></small></small>, <small><small>[[23/16]]</small></small>, <small>[[13/9]]</small>, '''[[29/20]]''', '''<u>[[16/11]]'''</u>, '''[[19/13]]''', [[22/15]], <small>[[25/17]]</small>, <small>[[28/19]]</small>, <small><small>[[31/21]]</small></small>, <small><small><small>[[34/23]]</small></small></small>
| ^^d5
| 44
|-
| 12
| 711.253
| '''[[3/2]]''', <small>[[32/21]]</small>, <small><small>[[29/19]]</small></small>, <small><small>[[26/17]]</small></small>, <small><small><small>[[23/15]]</small></small></small>
| ^5
| 48
|-
| 13
| 770.524
| <small><small><small>[[20/13]]</small></small></small>, <small>[[17/11]]</small>, [[31/20]], '''[[14/9]]''', '''<u>[[25/16]]'''</u>, [[11/7]], <small><small>[[30/19]]</small></small>, <small><small><small>[[19/12]]</small></small></small>
| ^^d6
| 52
|-
| 14
| 829.795
| <small><small><small>[[27/17]]</small></small></small>, <small>[[8/5]]</small>, '''<u>[[29/18]]'''</u>, '''<u>[[21/13]]'''</u>, '''<u>[[34/21]]'''</u>, [[13/8]], <small>[[31/19]]</small>, <small><small>[[18/11]]</small></small>
| ^m6
| 56
|-
| 15
| 889.067
| <small><small><small>[[23/14]]</small></small></small>, <small><small><small>[[28/17]]</small></small></small>, <small><small>[[33/20]]</small></small>, '''<u>[[5/3]]'''</u>, [[32/19]], <small>[[27/16]]</small>, <small><small>[[22/13]]</small></small>, <small><small><small>[[17/10]]</small></small></small>
| M6
| 60
|-
| 16
| 948.338
| <small><small>[[29/17]]</small></small>, <small>[[12/7]]</small>, '''[[31/18]]''', '''<u>[[19/11]]'''</u>, '''<u>[[26/15]]'''</u>, '''[[33/19]]''', <small><small>[[7/4]]</small></small>
| vA6, ^d7
| 64
|-
| 17
| 1007.609
| <small><small>[[30/17]]</small></small>, <small><small>[[23/13]]</small></small>, [[16/9]], '''<u>[[25/14]]'''</u>, '''<u>[[34/19]]'''</u>, [[9/5]], <small><small>[[29/16]]</small></small>, <small><small><small>[[20/11]]</small></small></small>
| m7
| 68
|-
| 18
| 1066.880
| <small><small><small>[[31/17]]</small></small></small>, <small>[[11/6]]</small>, '''[[24/13]]''', '''<u>[[13/7]]'''</u>, [[28/15]], <small><small>[[15/8]]</small></small>, <small><small><small>[[32/17]]</small></small></small>
| vM7
| 72
|-
| 19
| 1126.151
| <small><small><small>[[17/9]]</small></small></small>, <small>[[19/10]]</small>, '''[[21/11]]''', '''<u>[[23/12]]'''</u>, '''[[25/13]]''', [[27/14]], <small>[[29/15]]</small>, <small>[[31/16]]</small>, <small><small>[[33/17]]</small></small>
| vvA7
| 76
|-
| 20
| 1185.422
| [[2/1]]
| v1 +1 oct
| 80
|-
| 21
| 1244.693
| '''[[33/16]]''', [[31/15]], <small>[[29/14]]</small>, <small><small>[[27/13]]</small></small>, <small><small><small>[[25/12]]</small></small></small>
| vvA1 +1 oct
| 84
|-
| 22
| 1303.964
| <small><small><small>[[23/11]]</small></small></small>, <small>[[21/10]]</small>, [[19/9]], '''<u>[[17/8]]'''</u>, '''[[32/15]]''', <small>[[15/7]]</small>, <small><small>[[28/13]]</small></small>
| vm2 +1 oct
| 88
|-
| 23
| 1363.235
| <small><small>[[13/6]]</small></small>, [[24/11]], '''<u>[[11/5]]'''</u>, [[31/14]], <small>[[20/9]]</small>, <small><small><small>[[29/13]]</small></small></small>
| vvM2 +1 oct
| 92
|-
| 24
| 1422.507
| <small>[[9/4]]</small>, '''[[34/15]]''', '''<u>[[25/11]]'''</u>, '''[[16/7]]''', <small>[[23/10]]</small>, <small><small><small>[[30/13]]</small></small></small>
| ^^M2 +1 oct
| 96
|-
| 25
| 1481.778
| <small>[[7/3]]</small>, '''<u>[[33/14]]'''</u>, '''[[26/11]]''', <small>[[19/8]]</small>, <small><small>[[31/13]]</small></small>
| vvm3 +1 oct
| 100
|-
| 26
| 1541.049
| <small><small><small>[[12/5]]</small></small></small>, [[29/12]], '''<u>[[17/7]]'''</u>, '''[[22/9]]''', [[27/11]], <small>[[32/13]]</small>
| ^^m3 +1 oct
| 104
|-
| 27
| 1600.320
| [[5/2]], [[33/13]], <small>[[28/11]]</small>, <small><small>[[23/9]]</small></small>
| ^M3 +1 oct
| 108
|-
| 28
| 1659.591
| <small><small>[[18/7]]</small></small>, <small>[[31/12]]</small>, '''[[13/5]]''', '''<u>[[34/13]]'''</u>, [[21/8]], <small>[[29/11]]</small>
| ^^d4 +1 oct
| 112
|-
| 29
| 1718.862
| <small><small>[[8/3]]</small></small>, '''<u>[[27/10]]'''</u>, '''[[19/7]]''', <small>[[30/11]]</small>
| ^4 +1 oct
| 116
|-
| 30
| 1778.133
| <small><small><small>[[11/4]]</small></small></small>, '''[[25/9]]''', '''<u>[[14/5]]'''</u>, <small>[[31/11]]</small>, <small><small><small>[[17/6]]</small></small></small>
| A4 +1 oct
| 120
|-
| 31
| 1837.404
| <small><small>[[20/7]]</small></small>, '''[[23/8]]''', '''<u>[[26/9]]'''</u>, '''[[29/10]]''', [[32/11]]
| ^d5 +1 oct
| 124
|-
| 32
| 1896.675
| '''[[3/1]]'''
| P5 +1 oct
| 128
|-
| 33
| 1955.946
| '''<u>[[34/11]]'''</u>, '''<u>[[31/10]]'''</u>, '''[[28/9]]''', <small>[[25/8]]</small>, <small><small><small>[[22/7]]</small></small></small>
| vA5 +1 oct, ^d6 +1 oct
| 132
|-
| 34
| 2015.218
| <small>[[19/6]]</small>, '''<u>[[16/5]]'''</u>, [[29/9]], <small><small><small>[[13/4]]</small></small></small>
| m6 +1 oct
| 136
|-
| 35
| 2074.489
| <small>[[23/7]]</small>, '''[[33/10]]''', '''[[10/3]]'''
| vM6 +1 oct
| 140
|-
| 36
| 2133.760
| <small><small><small>[[27/8]]</small></small></small>, <small>[[17/5]]</small>, '''<u>[[24/7]]'''</u>, '''[[31/9]]'''
| vvA6 +1 oct
| 144
|-
| 37
| 2193.031
| <small><small>[[7/2]]</small></small>, '''<u>[[32/9]]'''</u>, [[25/7]], <small><small>[[18/5]]</small></small>
| vm7 +1 oct
| 148
|-
| 38
| 2252.302
| <small><small>[[29/8]]</small></small>, '''<u>[[11/3]]'''</u>, <small>[[26/7]]</small>
| vvM7 +1 oct
| 152
|-
| 39
| 2311.573
| <small><small>[[15/4]]</small></small>, [[34/9]], '''<u>[[19/5]]'''</u>, [[23/6]], <small><small><small>[[27/7]]</small></small></small>
| ^^M7 +1 oct
| 156
|-
| 40
| 2370.844
| <small><small><small>[[31/8]]</small></small></small>, <small><small><small>[[4/1]]</small></small></small>
| vv1 +2 oct
| 160
|-
| 41
| 2430.115
| <small><small>[[33/8]]</small></small>
| ^^1 +2 oct
| 164
|-
| 42
| 2489.386
| <small><small><small>[[29/7]]</small></small></small>, <small>[[25/6]]</small>, '''<u>[[21/5]]'''</u>, <small>[[17/4]]</small>
| vvm2 +2 oct
| 168
|-
| 43
| 2548.658
| <small><small><small>[[30/7]]</small></small></small>, [[13/3]], <small>[[22/5]]</small>, <small><small><small>[[31/7]]</small></small></small>
| ^^m2 +2 oct
| 172
|-
| 44
| 2607.929
| '''<u>[[9/2]]'''</u>, <small><small>[[32/7]]</small></small>
| ^M2 +2 oct
| 176
|-
| 45
| 2667.200
| <small><small><small>[[23/5]]</small></small></small>, '''<u>[[14/3]]'''</u>, <small>[[33/7]]</small>
| ^^d3 +2 oct
| 180
|-
| 46
| 2726.471
| <small><small><small>[[19/4]]</small></small></small>, [[24/5]], '''<u>[[29/6]]'''</u>, '''[[34/7]]'''
| ^m3 +2 oct
| 184
|-
| 47
| 2785.742
| '''<u>[[5/1]]'''</u>
| M3 +2 oct
| 188
|-
| 48
| 2845.013
| '''<u>[[31/6]]'''</u>, '''[[26/5]]''', <small><small><small>[[21/4]]</small></small></small>
| vA3 +2 oct, ^d4 +2 oct
| 192
|-
| 49
| 2904.284
| '''[[16/3]]''', <small>[[27/5]]</small>
| P4 +2 oct
| 196
|-
| 50
| 2963.555
| [[11/2]], <small>[[28/5]]</small>
| vA4 +2 oct
| 200
|-
| 51
| 3022.826
| <small><small>[[17/3]]</small></small>, '''[[23/4]]''', <small><small>[[29/5]]</small></small>
| d5 +2 oct
| 204
|-
| 52
| 3082.097
| <small><small>[[6/1]]</small></small>
| v5 +2 oct
| 208
|-
| 53
| 3141.369
| <small>[[31/5]]</small>
| vvA5 +2 oct
| 212
|-
| 54
| 3200.640
| <small><small><small>[[25/4]]</small></small></small>, '''[[19/3]]''', [[32/5]]
| vm6 +2 oct
| 216
|-
| 55
| 3259.911
| <small>[[13/2]]</small>, '''[[33/5]]''', <small><small>[[20/3]]</small></small>
| vvM6 +2 oct
| 220
|-
| 56
| 3319.182
| [[27/4]], '''<u>[[34/5]]'''</u>
| ^^M6 +2 oct
| 224
|-
| 57
| 3378.453
| '''[[7/1]]'''
| vvm7 +2 oct
| 228
|-
| 58
| 3437.724
| '''[[29/4]]''', [[22/3]]
| ^^m7 +2 oct
| 232
|-
| 59
| 3496.995
| '''[[15/2]]''', <small><small><small>[[23/3]]</small></small></small>
| ^M7 +2 oct
| 236
|-
| 60
| 3556.266
| [[31/4]]
| ^^d1 +3 oct
| 240
|-
| 61
| 3615.537
| <small>[[8/1]]</small>
| ^1 +3 oct
| 244
|-
| 62
| 3674.809
| <small><small>[[33/4]]</small></small>, '''<u>[[25/3]]'''</u>
| ^^d2 +3 oct
| 248
|-
| 63
| 3734.080
| <small><small><small>[[17/2]]</small></small></small>, '''<u>[[26/3]]'''</u>
| ^m2 +3 oct
| 252
|-
| 64
| 3793.351
| [[9/1]]
| M2 +3 oct
| 256
|-
| 65
| 3852.622
| [[28/3]]
| vA2 +3 oct, ^d3 +3 oct
| 260
|-
| 66
| 3911.893
| [[19/2]], <small>[[29/3]]</small>
| m3 +3 oct
| 264
|-
| 67
| 3971.164
| <small>[[10/1]]</small>
| vM3 +3 oct
| 268
|-
| 68
| 4030.435
| [[31/3]]
| vvA3 +3 oct
| 272
|-
| 69
| 4089.706
| <small>[[21/2]]</small>, '''[[32/3]]'''
| v4 +3 oct
| 276
|-
| 70
| 4148.977
| '''<u>[[11/1]]'''</u>
| vvA4 +3 oct
| 280
|-
| 71
| 4208.248
| '''[[34/3]]''', <small><small>[[23/2]]</small></small>
| vd5 +3 oct
| 284
|-
| 72
| 4267.520
|
| vv5 +3 oct
| 288
|-
| 73
| 4326.791
| <small><small><small>[[12/1]]</small></small></small>
| ^^5 +3 oct
| 292
|-
| 74
| 4386.062
| [[25/2]]
| vvm6 +3 oct
| 296
|-
| 75
| 4445.333
| '''<u>[[13/1]]'''</u>
| ^^m6 +3 oct
| 300
|-
| 76
| 4504.604
| '''<u>[[27/2]]'''</u>
| ^M6 +3 oct
| 304
|-
| 77
| 4563.875
| '''[[14/1]]'''
| ^^d7 +3 oct
| 308
|-
| 78
| 4623.146
| '''[[29/2]]'''
| ^m7 +3 oct
| 312
|-
| 79
| 4682.417
| '''[[15/1]]'''
| M7 +3 oct
| 316
|-
| 80
| 4741.688
| '''<u>[[31/2]]'''</u>
| vA7 +3 oct, ^d1 +4 oct
| 320
|-
| 81
| 4800.959
| '''<u>[[16/1]]'''</u>
| P1 +4 oct
| 324
|-
| 82
| 4860.231
| '''[[33/2]]'''
| vA1 +4 oct, ^d2 +4 oct
| 328
|-
| 83
| 4919.502
| [[17/1]]
| m2 +4 oct
| 332
|-
| 84
| 4978.773
| <small><small><small>[[18/1]]</small></small></small>
| vM2 +4 oct
| 336
|-
| 85
| 5038.044
|
| vvA2 +4 oct
| 340
|-
| 86
| 5097.315
| '''<u>[[19/1]]'''</u>
| vm3 +4 oct
| 344
|-
| 87
| 5156.586
|
| vvM3 +4 oct
| 348
|-
| 88
| 5215.857
| <small><small><small>[[20/1]]</small></small></small>
| ^^M3 +4 oct
| 352
|-
| 89
| 5275.128
| '''<u>[[21/1]]'''</u>
| vv4 +4 oct
| 356
|-
| 90
| 5334.399
| <small>[[22/1]]</small>
| ^^4 +4 oct
| 360
|-
| 91
| 5393.671
|
| ^A4 +4 oct
| 364
|-
| 92
| 5452.942
| <small><small>[[23/1]]</small></small>
| ^^d5 +4 oct
| 368
|-
| 93
| 5512.213
| [[24/1]]
| ^5 +4 oct
| 372
|-
| 94
| 5571.484
| '''<u>[[25/1]]'''</u>
| ^^d6 +4 oct
| 376
|-
| 95
| 5630.755
| '''[[26/1]]'''
| ^m6 +4 oct
| 380
|-
| 96
| 5690.026
| <small>[[27/1]]</small>
| M6 +4 oct
| 384
|-
| 97
| 5749.297
| <small>[[28/1]]</small>
| vA6 +4 oct, ^d7 +4 oct
| 388
|-
| 98
| 5808.568
| <small><small>[[29/1]]</small></small>
| m7 +4 oct
| 392
|-
| 99
| 5867.839
| <small><small>[[30/1]]</small></small>
| vM7 +4 oct
| 396
|-
| 100
| 5927.110
| <small>[[31/1]]</small>
| vvA7 +4 oct
| 400
|-
| 101
| 5986.382
| [[32/1]]
| v1 +5 oct
| 404
|-
| 102
| 6045.653
| '''[[33/1]]'''
| vvA1 +5 oct
| 408
|-
| 103
| 6104.924
| '''<u>[[34/1]]'''</u>
| vm2 +5 oct
| 412
|}


71zpi is distinguished by its extensive [[EDO-span|EDO-deviation]] and substantial zeta strength, qualifying it as a strong candidate for no-octave tuning systems. It is noteworthy that only [[19zpi]] exhibits both a greater octave error and stronger zeta height and integral than 71zpi, although 71zpi still has a more pronounced zeta gap. Other notable [[Zeta peak index|zeta peak indexes]] in this category include [[61zpi]], [[84zpi]], [[110zpi]], [[137zpi]], [[151zpi]], [[222zpi]], and [[273zpi]], each demonstrating characteristics that make them suitable for similar applications.
=== Approximation to JI in 71zpi with prime 2 removed ===


=== Harmonic series ===
==== Interval mappings in 71zpi with prime 2 removed ====
{{Harmonics in cet|59.3329806724710|columns=15|title=Approximation of harmonics in 71zpi}}
{{Harmonics in cet|59.3329806724710|columns=17|start=16|title=Approximation of harmonics in 71zpi}}


== Intervals ==
The following tables show how 34-integer-limit intervals are represented in 71zpi with prime 2 removed. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italics''.
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.
{| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"
 
|+ style="white-space: nowrap;" | 34-integer-limit intervals in 71zpi with prime 2 removed (by direct approximation)
{{todo|Fill in the blank sections with the ups and downs notation from 20-EDO and 182-EDO.|inline=1|comment=Additionally, incorporate a new column adjacent to each notation to indicate the corresponding octave.}}
|-
 
! Ratio
{| class="wikitable center-all right-2 left-3"
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
| [[34/1]]
| -0.032
| -0.053
|-
| [[34/19]]
| +0.166
| +0.281
|- style="background-color: #cccccc;"
| ''[[23/12]]''
| ''-0.168''
| ''-0.284''
|-
| '''[[19/1]]'''
| '''-0.198'''
| '''-0.334'''
|-
| [[14/3]]
| +0.329
| +0.555
|-
| [[19/5]]
| +0.374
| +0.631
|-
| [[21/13]]
| -0.458
| -0.772
|-
| [[34/5]]
| +0.540
| +0.911
|-
| '''[[5/1]]'''
| '''-0.572'''
| '''-0.965'''
|-
| [[30/29]]
| +0.580
| +0.978
|- style="background-color: #cccccc;"
| ''[[24/7]]''
| ''+0.631''
| ''+1.064''
|-
| [[27/10]]
| -0.689
| -1.163
|-
| [[26/9]]
| +0.787
| +1.327
|-
| [[15/14]]
| -0.901
| -1.519
|-
| [[25/19]]
| -0.946
| -1.595
|- style="background-color: #cccccc;"
| ''[[16/1]]''
| ''+0.959''
| ''+1.619''
|- style="background-color: #cccccc;"
| ''[[17/8]]''
| ''-0.991''
| ''-1.672''
|-
| [[31/22]]
| -1.007
| -1.698
|-
| [[27/22]]
| +1.080
| +1.821
|-
| [[34/25]]
| +1.112
| +1.876
|-
| [[25/1]]
| -1.144
| -1.929
|-
| [[29/6]]
| -1.151
| -1.943
|- style="background-color: #cccccc;"
| ''[[19/16]]''
| ''-1.157''
| ''-1.953''
|-
| [[25/11]]
| +1.197
| +2.020
|-
| [[27/2]]
| -1.261
| -2.128
|-
| [[29/28]]
| -1.480
| -2.497
|- style="background-color: #cccccc;"
| ''[[16/5]]''
| ''+1.531''
| ''+2.584''
|-
| [[31/28]]
| +1.604
| +2.706
|-
| [[11/5]]
| -1.769
| -2.984
|-
| [[31/6]]
| +1.932
| +3.260
|-
| [[31/27]]
| -2.086
| -3.520
|- style="background-color: #cccccc;"
| ''[[25/16]]''
| ''-2.103''
| ''-3.548''
|-
| [[19/11]]
| +2.143
| +3.615
|-
| [[33/26]]
| +2.152
| +3.632
|- style="background-color: #cccccc;"
| ''[[32/27]]''
| ''+2.221''
| ''+3.746''
|-
| [[34/11]]
| +2.309
| +3.896
|-
| '''[[11/1]]'''
| '''-2.341'''
| '''-3.949'''
|-
| [[31/30]]
| +2.504
| +4.225
|-
| [[14/11]]
| -2.610
| -4.404
|-
| [[33/14]]
| -2.669
| -4.504
|-
| [[31/10]]
| -2.775
| -4.683
|-
| [[11/3]]
| +2.939
| +4.959
|- style="background-color: #cccccc;"
| ''[[32/9]]''
| ''-3.059''
| ''-5.161''
|-
| [[31/29]]
| +3.084
| +5.203
|- style="background-color: #cccccc;"
| ''[[16/11]]''
| ''+3.300''
| ''+5.568''
|-
| [[31/2]]
| -3.347
| -5.647
|- style="background-color: #cccccc;"
| ''[[21/16]]''
| ''+3.388''
| ''+5.716''
|-
| [[15/11]]
| -3.511
| -5.923
|-
| [[28/27]]
| -3.690
| -6.225
|-
| [[25/14]]
| +3.807
| +6.423
|- style="background-color: #cccccc;"
| ''[[16/13]]''
| ''-3.846''
| ''-6.488''
|-
| [[26/15]]
| -3.921
| -6.616
|-
| [[9/2]]
| +4.019
| +6.780
|-
| [[29/22]]
| -4.090
| -6.901
|-
| [[29/18]]
| +4.128
| +6.965
|-
| [[25/3]]
| +4.136
| +6.978
|- style="background-color: #cccccc;"
| ''[[24/17]]''
| ''-4.289''
| ''-7.235''
|- style="background-color: #cccccc;"
| ''[[32/31]]''
| ''+4.307''
| ''+7.266''
|-
| [[21/1]]
| +4.347
| +7.335
|-
| [[34/21]]
| -4.379
| -7.388
|-
| [[14/5]]
| -4.379
| -7.388
|-
| [[26/3]]
| -4.493
| -7.581
|-
| [[21/19]]
| +4.545
| +7.669
|-
| [[10/9]]
| -4.590
| -7.745
|-
| [[5/3]]
| +4.708
| +7.943
|-
| [[19/14]]
| +4.753
| +8.019
|-
| '''[[13/1]]'''
| '''+4.805'''
| '''+8.107'''
|-
| [[13/7]]
| -4.822
| -8.135
|-
| [[34/13]]
| -4.837
| -8.160
|- style="background-color: #cccccc;"
| ''[[23/20]]''
| ''-4.876''
| ''-8.227''
|-
| [[21/5]]
| +4.919
| +8.300
|-
| [[17/7]]
| +4.919
| +8.300
|-
| [[14/1]]
| -4.951
| -8.353
|-
| [[19/13]]
| -5.003
| -8.441
|-
| [[19/3]]
| +5.082
| +8.574
|-
| [[29/27]]
| -5.170
| -8.723
|-
| [[34/3]]
| +5.248
| +8.854
|-
| '''[[3/1]]'''
| '''-5.280'''
| '''-8.908'''
|-
| [[13/5]]
| +5.377
| +9.072
|- style="background-color: #cccccc;"
| ''[[23/4]]''
| ''-5.448''
| ''-9.192''
|- style="background-color: #cccccc;"
| ''[[24/13]]''
| ''+5.453''
| ''+9.199''
|-
| [[25/21]]
| -5.491
| -9.264
|-
| [[14/9]]
| +5.608
| +9.462
|-
| [[19/15]]
| +5.653
| +9.538
|-
| [[34/15]]
| +5.820
| +9.819
|-
| [[15/1]]
| -5.851
| -9.872
|-
| [[29/10]]
| -5.859
| -9.885
|- style="background-color: #cccccc;"
| ''[[8/7]]''
| ''+5.910''
| ''+9.972''
|-
| [[25/13]]
| -5.949
| -10.037
|- style="background-color: #cccccc;"
| ''[[33/32]]''
| ''+5.998''
| ''+10.120''
|-
| [[27/26]]
| -6.066
| -10.235
|- style="background-color: #cccccc;"
| ''[[16/3]]''
| ''+6.239''
| ''+10.526''
|-
| [[22/9]]
| -6.359
| -10.729
|-
| [[29/2]]
| -6.431
| -10.850
|-
| [[33/25]]
| -6.477
| -10.927
|-
| [[21/11]]
| +6.688
| +11.284
|- style="background-color: #cccccc;"
| ''[[16/15]]''
| ''+6.811''
| ''+11.491''
|-
| [[33/2]]
| +6.958
| +11.739
|-
| [[33/5]]
| -7.048
| -11.892
|-
| [[13/11]]
| +7.146
| +12.056
|-
| [[31/18]]
| +7.212
| +12.168
|-
| [[31/9]]
| -7.366
| -12.427
|- style="background-color: #cccccc;"
| ''[[32/29]]''
| ''+7.391''
| ''+12.469''
|-
| [[33/19]]
| -7.422
| -12.523
|-
| [[26/11]]
| -7.432
| -12.539
|-
| [[33/10]]
| +7.529
| +12.703
|-
| [[34/33]]
| +7.589
| +12.803
|-
| [[33/1]]
| -7.620
| -12.857
|- style="background-color: #cccccc;"
| ''[[32/15]]''
| ''-7.767''
| ''-13.104''
|-
| [[29/4]]
| +8.147
| +13.745
|-
| [[31/26]]
| -8.152
| -13.754
|-
| [[11/9]]
| +8.219
| +13.866
|- style="background-color: #cccccc;"
| ''[[32/3]]''
| ''-8.339''
| ''-14.069''
|- style="background-color: #cccccc;"
| ''[[33/16]]''
| ''-8.580''
| ''-14.475''
|-
| [[26/25]]
| -8.629
| -14.559
|- style="background-color: #cccccc;"
| ''[[16/7]]''
| ''-8.668''
| ''-14.624''
|-
| [[29/20]]
| +8.719
| +14.710
|-
| [[15/2]]
| +8.726
| +14.723
|-
| [[28/9]]
| -8.969
| -15.133
|- style="background-color: #cccccc;"
| ''[[23/8]]''
| ''+9.130''
| ''+15.404''
|-
| [[26/5]]
| -9.201
| -15.523
|-
| [[3/2]]
| +9.298
| +15.688
|-
| [[25/9]]
| +9.416
| +15.886
|- style="background-color: #cccccc;"
| ''[[23/18]]''
| ''-9.467''
| ''-15.972''
|-
| [[26/19]]
| -9.575
| -16.154
|-
| '''[[7/1]]'''
| '''+9.627'''
| '''+16.242'''
|-
| [[34/7]]
| -9.659
| -16.296
|-
| [[17/13]]
| +9.741
| +16.435
|-
| [[14/13]]
| -9.756
| -16.460
|-
| [[26/1]]
| -9.773
| -16.488
|-
| [[19/7]]
| -9.825
| -16.576
|-
| [[10/3]]
| -9.870
| -16.652
|-
| [[9/5]]
| -9.988
| -16.851
|-
| [[13/3]]
| +10.085
| +17.015
|-
| [[23/17]]
| +10.121
| +17.076
|-
| [[7/5]]
| +10.199
| +17.207
|-
| [[21/17]]
| -10.199
| -17.207
|- style="background-color: #cccccc;"
| ''[[24/1]]''
| ''+10.258''
| ''+17.307''
|- style="background-color: #cccccc;"
| ''[[17/12]]''
| ''-10.289''
| ''-17.360''
|-
| [[33/31]]
| +10.305
| +17.386
|-
| [[19/9]]
| +10.361
| +17.481
|-
| [[29/9]]
| -10.450
| -17.630
|- style="background-color: #cccccc;"
| ''[[24/19]]''
| ''+10.456''
| ''+17.641''
|-
| [[34/9]]
| +10.528
| +17.762
|-
| [[9/1]]
| -10.559
| -17.815
|-
| [[15/13]]
| -10.657
| -17.979
|- style="background-color: #cccccc;"
| ''[[13/8]]''
| ''-10.732''
| ''-18.107''
|-
| [[25/7]]
| -10.771
| -18.172
|- style="background-color: #cccccc;"
| ''[[24/5]]''
| ''+10.830''
| ''+18.271''
|-
| [[27/14]]
| -10.888
| -18.370
|-
| [[22/15]]
| -11.067
| -18.672
|- style="background-color: #cccccc;"
| ''[[21/8]]''
| ''-11.190''
| ''-18.879''
|-
| [[31/4]]
| +11.231
| +18.948
|-
| [[29/26]]
| -11.236
| -18.957
|- style="background-color: #cccccc;"
| ''[[32/11]]''
| ''-11.278''
| ''-19.027''
|- style="background-color: #cccccc;"
| ''[[25/24]]''
| ''-11.401''
| ''-19.236''
|- style="background-color: #cccccc;"
| ''[[16/9]]''
| ''+11.519''
| ''+19.434''
|-
| [[22/3]]
| -11.639
| -19.637
|-
| [[31/20]]
| +11.803
| +19.913
|-
| [[33/28]]
| +11.908
| +20.092
|-
| [[11/7]]
| -11.968
| -20.191
|-
| [[31/15]]
| -12.074
| -20.370
|-
| [[11/2]]
| +12.237
| +20.646
|-
| [[33/13]]
| -12.425
| -20.964
|- style="background-color: #cccccc;"
| ''[[32/25]]''
| ''-12.475''
| ''-21.047''
|- style="background-color: #cccccc;"
| ''[[24/11]]''
| ''+12.598''
| ''+21.255''
|-
| [[31/3]]
| -12.645
| -21.335
|-
| [[11/10]]
| +12.809
| +21.611
|-
| [[31/14]]
| -12.974
| -21.890
|- style="background-color: #cccccc;"
| ''[[32/5]]''
| ''-13.047''
| ''-22.012''
|-
| [[27/4]]
| +13.317
| +22.468
|-
| [[33/29]]
| +13.389
| +22.589
|- style="background-color: #cccccc;"
| ''[[32/19]]''
| ''-13.420''
| ''-22.642''
|-
| [[29/12]]
| +13.427
| +22.653
|-
| [[25/2]]
| +13.434
| +22.666
|-
| [[27/11]]
| -13.498
| -22.774
|- style="background-color: #cccccc;"
| ''[[17/16]]''
| ''+13.587''
| ''+22.923''
|- style="background-color: #cccccc;"
| ''[[29/23]]''
| ''+13.595''
| ''+22.937''
|- style="background-color: #cccccc;"
| ''[[32/1]]''
| ''-13.618''
| ''-22.976''
|-
| [[28/15]]
| -13.677
| -23.076
|-
| [[27/20]]
| +13.889
| +23.432
|-
| [[5/2]]
| +14.006
| +23.631
|-
| [[26/21]]
| -14.120
| -23.823
|- style="background-color: #cccccc;"
| ''[[30/23]]''
| ''+14.174''
| ''+23.915''
|-
| [[28/3]]
| -14.249
| -24.041
|-
| [[19/2]]
| +14.380
| +24.261
|- style="background-color: #cccccc;"
| ''[[24/23]]''
| ''-14.410''
| ''-24.311''
|-
| '''[[17/1]]'''
| '''+14.546'''
| '''+24.542'''
|-
| '''[[2/1]]'''
| '''-14.578'''
| '''-24.595'''
|-
| [[27/25]]
| -14.695
| -24.793
|-
| [[19/17]]
| -14.744
| -24.876
|- style="background-color: #cccccc;"
| ''[[23/6]]''
| ''-14.746''
| ''-24.879''
|-
| [[7/3]]
| +14.907
| +25.150
|-
| [[19/10]]
| +14.952
| +25.226
|- style="background-color: #cccccc;"
| ''[[20/17]]''
| ''+14.997''
| ''+25.303''
|-
| [[23/7]]
| +15.040
| +25.375
|- style="background-color: #cccccc;"
| ''[[28/23]]''
| ''+15.075''
| ''+25.434''
|-
| [[17/5]]
| +15.118
| +25.507
|-
| [[10/1]]
| -15.150
| -25.560
|-
| [[29/15]]
| -15.157
| -25.573
|- style="background-color: #cccccc;"
| ''[[12/7]]''
| ''+15.209''
| ''+25.659''
|-
| [[27/5]]
| -15.267
| -25.758
|-
| [[13/9]]
| +15.364
| +25.922
|-
| [[15/7]]
| -15.478
| -26.115
|- style="background-color: #cccccc;"
| ''[[8/1]]''
| ''+15.537''
| ''+26.214''
|- style="background-color: #cccccc;"
| ''[[17/4]]''
| ''-15.569''
| ''-26.267''
|-
| [[31/11]]
| -15.584
| -26.294
|-
| [[27/19]]
| -15.641
| -26.389
|-
| [[25/17]]
| -15.690
| -26.471
|-
| [[29/3]]
| -15.729
| -26.538
|- style="background-color: #cccccc;"
| ''[[19/8]]''
| ''-15.735''
| ''-26.548''
|-
| [[25/22]]
| +15.775
| +26.615
|-
| [[34/27]]
| +15.807
| +26.670
|-
| [[27/1]]
| -15.839
| -26.723
|-
| [[29/14]]
| -16.058
| -27.093
|- style="background-color: #cccccc;"
| ''[[8/5]]''
| ''+16.109''
| ''+27.179''
|-
| [[22/5]]
| -16.347
| -27.580
|-
| [[31/12]]
| +16.510
| +27.856
|- style="background-color: #cccccc;"
| ''[[31/23]]''
| ''+16.679''
| ''+28.140''
|- style="background-color: #cccccc;"
| ''[[25/8]]''
| ''-16.681''
| ''-28.144''
|-
| [[22/19]]
| -16.721
| -28.210
|-
| [[31/25]]
| -16.782
| -28.313
|- style="background-color: #cccccc;"
| ''[[27/16]]''
| ''-16.798''
| ''-28.342''
|-
| [[17/11]]
| +16.887
| +28.491
|-
| [[22/1]]
| -16.918
| -28.544
|-
| [[28/11]]
| -17.188
| -28.999
|-
| [[33/7]]
| -17.247
| -29.099
|-
| [[31/5]]
| -17.353
| -29.278
|-
| [[11/6]]
| +17.517
| +29.554
|- style="background-color: #cccccc;"
| ''[[23/22]]''
| ''-17.685''
| ''-29.838''
|-
| [[31/19]]
| -17.727
| -29.908
|- style="background-color: #cccccc;"
| ''[[11/8]]''
| ''-17.878''
| ''-30.163''
|-
| [[34/31]]
| +17.893
| +30.189
|-
| '''[[31/1]]'''
| '''-17.925'''
| '''-30.243'''
|- style="background-color: #cccccc;"
| ''[[32/21]]''
| ''-17.966''
| ''-30.311''
|-
| [[30/11]]
| -18.089
| -30.519
|-
| [[28/25]]
| -18.385
| -31.019
|- style="background-color: #cccccc;"
| ''[[32/13]]''
| ''-18.424''
| ''-31.084''
|-
| [[9/4]]
| +18.597
| +31.375
|-
| [[29/11]]
| -18.668
| -31.496
|-
| [[25/6]]
| +18.714
| +31.574
|- style="background-color: #cccccc;"
| ''[[27/23]]''
| ''+18.765''
| ''+31.659''
|- style="background-color: #cccccc;"
| ''[[31/16]]''
| ''-18.885''
| ''-31.861''
|-
| [[21/2]]
| +18.925
| +31.930
|-
| [[28/5]]
| -18.957
| -31.983
|-
| [[20/9]]
| -19.168
| -32.340
|-
| [[6/5]]
| -19.286
| -32.538
|-
| [[28/19]]
| -19.331
| -32.614
|-
| [[13/2]]
| +19.383
| +32.702
|-
| [[26/7]]
| -19.400
| -32.731
|- style="background-color: #cccccc;"
| ''[[23/10]]''
| ''-19.454''
| ''-32.822''
|-
| [[21/10]]
| +19.497
| +32.895
|-
| [[17/14]]
| +19.497
| +32.895
|-
| [[28/1]]
| -19.529
| -32.948
|- style="background-color: #cccccc;"
| ''[[18/17]]''
| ''+19.588''
| ''+33.047''
|-
| [[19/6]]
| +19.660
| +33.169
|-
| [[17/3]]
| +19.826
| +33.450
|-
| [[6/1]]
| -19.858
| -33.503
|-
| [[23/13]]
| +19.862
| +33.511
|-
| [[29/25]]
| -19.865
| -33.516
|- style="background-color: #cccccc;"
| ''[[20/7]]''
| ''+19.916''
| ''+33.602''
|-
| [[13/10]]
| +19.955
| +33.667
|- style="background-color: #cccccc;"
| ''[[23/2]]''
| ''-20.026''
| ''-33.787''
|- style="background-color: #cccccc;"
| ''[[13/12]]''
| ''-20.030''
| ''-33.795''
|-
| [[9/7]]
| -20.186
| -34.058
|-
|-
!Step
| [[30/19]]
!Cents
| -20.231
!Ratios
| -34.134
! colspan="3" |[[Ups and Downs Notation]] from [[20edo|20EDO]]
! colspan="3" |[[Ups and Downs Notation]] from [[182edo|182EDO]]
|-
|-
|0
| [[23/21]]
|0.000
| +20.320
|[[1/1]]
| +34.283
|unison
|P1
|D
|unison
|P1
|D
|-
|-
|1
| [[17/15]]
|59.333
| +20.398
|[[30/29]], [[29/28]]
| +34.414
|up unison, upminor 2nd
|^1, ^m2
|^D, ^Eb
|
|
|
|-
|-
|2
| [[30/1]]
|118.666
| -20.429
|[[15/14]]
| -34.468
|dup unison, mid 2nd
|^^1, ~2
|^^D, vvE
|
|
|
|-
|-
|3
| [[29/5]]
|177.999
| -20.437
|[[10/9]]
| -34.481
|downmajor 2nd
|- style="background-color: #cccccc;"
|vM2
| ''[[7/4]]''
|vE
| ''-20.488''
|
| ''-34.567''
|
|
|-
|-
|4
| [[27/13]]
|237.332
| -20.644
|[[8/7]]
| -34.830
|major 2nd, minor 3rd
|M2, m3
|E, F
|
|
|
|-
|-
|5
| [[29/19]]
|296.665
| -20.811
|[[13/11]], [[19/16]], [[6/5]]
| -35.111
|upminor 3rd
|- style="background-color: #cccccc;"
|^m3
| ''[[8/3]]''
|^F
| ''+20.817''
|
| ''+35.122''
|
|
|-
|-
|6
| [[34/29]]
|355.998
| +20.977
|[[11/9]], [[27/22]], [[16/13]]
| +35.392
|mid 3rd
|- style="background-color: #cccccc;"
|~3
| ''[[32/23]]''
|^^F, vvF#
| ''+20.985''
|
| ''+35.406''
|
|
|-
|-
|7
| '''[[29/1]]'''
|415.331
| '''-21.009'''
|[[5/4]], [[14/11]]
| '''-35.445'''
|downmajor 3rd
|vM3
|vF#
|
|
|
|-
|-
|8
| [[22/21]]
|474.664
| -21.266
|[[25/19]], [[4/3]]
| -35.879
|major 3rd, perfect fourth
|- style="background-color: #cccccc;"
|M3, P4
| ''[[15/8]]''
|F#, G
| ''-21.389''
|
| ''-36.086''
|
|
|-
|-
|9
| [[33/4]]
|533.997
| +21.536
|[[15/11]]
| +36.334
|up-fourth
|^4
|^G
|
|
|
|-
|-
|10
| [[22/13]]
|593.330
| -21.724
|[[7/5]], [[31/22]]
| -36.651
|mid fourth, mid fifth
|- style="background-color: #cccccc;"
|~4, ~5
| ''[[29/16]]''
|^^G, vvA
| ''-21.968''
|
| ''-37.064''
|
|
|-
|-
|11
| [[33/20]]
|652.663
| +22.107
|[[16/11]], [[19/13]]
| +37.299
|down-fifth
|v5
|vA
|
|
|
|-
|-
|12
| [[33/17]]
|711.996
| -22.167
|[[3/2]]
| -37.399
|fifth
|P5, m6
|A
|
|
|
|-
|-
|13
| [[31/21]]
|771.329
| -22.273
|[[14/9]], [[25/16]], [[11/7]]
| -37.577
|upfifth, upminor 6th
|^5, ^m6
|^A, ^Bb
|
|
|
|-
|-
|14
| [[29/8]]
|830.662
| +22.725
|[[8/5]], [[21/13]], [[13/8]]
| +38.340
|mid 6th
|~6
|^^A, vvB
|
|
|
|-
|-
|15
| [[31/13]]
|889.995
| -22.730
|[[5/3]]
| -38.350
|downmajor 6th
|vM6
|vB
|
|
|
|-
|-
|16
| [[18/11]]
|949.328
| -22.797
|[[19/11]], [[26/15]], [[7/4]]
| -38.462
|major 6th, minor 7th
|- style="background-color: #cccccc;"
|M6, m7
| ''[[33/8]]''
|B, C
| ''-23.158''
|
| ''-39.071''
|
|- style="background-color: #cccccc;"
|
| ''[[32/7]]''
| ''-23.245''
| ''-39.219''
|-
|-
|17
| [[15/4]]
|1008.661
| +23.304
|[[9/5]]
| +39.318
|upminor 7th
|- style="background-color: #cccccc;"
|^m7
| ''[[23/16]]''
|^C
| ''+23.708''
|
| ''+39.999''
|
|- style="background-color: #cccccc;"
|
| ''[[29/17]]''
| ''+23.716''
| ''+40.013''
|-
|-
|18
| [[4/3]]
|1067.994
| -23.876
|[[13/7]]
| -40.283
|mid 7th
|~7
|^^C, vvD
|
|
|
|-
|-
|19
| [[25/18]]
|1127.327
| +23.994
|[[23/12]]
| +40.481
|downmajor 7th
|- style="background-color: #cccccc;"
|vM7
| ''[[23/9]]''
|vD
| ''-24.045''
|
| ''-40.567''
|
|
|-
|-
|20
| [[7/2]]
|1186.660
| +24.205
|[[2/1]]
| +40.838
|octave
|- style="background-color: #cccccc;"
|P8
| ''[[30/17]]''
|D
| ''+24.295''
|
| ''+40.990''
|
|
|-
|-
|22
| [[26/17]]
|1305.326
| -24.319
|[[17/8]]
| -41.030
|
|
|
|
|
|
|-
|-
|23
| [[28/13]]
|1364.659
| -24.334
|[[11/5]]
| -41.055
|
|
|
|
|
|
|-
|-
|25
| [[20/3]]
|1483.325
| -24.448
|[[7/3]]
| -41.248
|
|- style="background-color: #cccccc;"
|
| ''[[18/7]]''
|
| ''+24.507''
|
| ''+41.347''
|
|
|-
|-
|27
| [[18/5]]
|1601.990
| -24.565
|[[5/2]]
| -41.446
|
|
|
|
|
|
|-
|-
|28
| [[13/6]]
|1661.323
| +24.663
|[[13/5]]
| +41.610
|
|
|
|
|
|
|-
|-
|29
| '''[[23/1]]'''
|1720.656
| '''+24.667'''
|[[8/3]], [[27/10]]
| '''+41.618'''
|
|
|
|
|
|
|-
|-
|30
| [[34/23]]
|1779.989
| -24.699
|[[14/5]]
| -41.671
|
|- style="background-color: #cccccc;"
|
| ''[[20/13]]''
|
| ''+24.738''
|
| ''+41.738''
|
|
|-
|-
|32
| [[10/7]]
|1898.655
| -24.777
|[[3/1]]
| -41.802
|
|- style="background-color: #cccccc;"
|
| ''[[26/23]]''
|
| ''+24.831''
|
| ''+41.894''
|
|- style="background-color: #cccccc;"
|
| ''[[12/1]]''
| ''+24.836''
| ''+41.902''
|-
|-
|33
| [[23/19]]
|1957.988
| +24.865
|[[31/10]]
| +41.952
|
|- style="background-color: #cccccc;"
|
| ''[[17/6]]''
|
| ''-24.867''
|
| ''-41.955''
|
|
|-
|-
|34
| [[19/18]]
|2017.321
| +24.939
|[[16/5]]
| +42.076
|
|- style="background-color: #cccccc;"
|
| ''[[19/12]]''
|
| ''-25.034''
|
| ''-42.236''
|
|
|-
|-
|35
| [[17/9]]
|2076.654
| +25.106
|[[10/3]]
| +42.357
|
|
|
|
|
|
|-
|-
|36
| [[18/1]]
|2135.987
| -25.137
|[[24/7]]
| -42.411
|
|- style="background-color: #cccccc;"
|
| ''[[28/17]]''
|
| ''+25.196''
|
| ''+42.510''
|
|- style="background-color: #cccccc;"
|
| ''[[21/20]]''
| ''-25.196''
| ''-42.510''
|-
|-
|37
| [[30/13]]
|2195.320
| -25.235
|[[7/2]], [[32/9]]
| -42.575
|
|
|
|
|
|
|-
|-
|38
| [[23/5]]
|2254.653
| +25.239
|[[11/3]]
| +42.582
|
|- style="background-color: #cccccc;"
|
| ''[[13/4]]''
|
| ''-25.310''
|
| ''-42.702''
|
|
|-
|-
|39
| [[29/21]]
|2313.986
| -25.356
|[[19/5]]
| -42.780
|
|- style="background-color: #cccccc;"
|
| ''[[12/5]]''
|
| ''+25.407''
|
| ''+42.866''
|
|
|-
|-
|40
| [[27/7]]
|2373.319
| -25.466
|[[4/1]]
| -42.965
|
|- style="background-color: #cccccc;"
|
| ''[[21/4]]''
|
| ''-25.768''
|
| ''-43.475''
|
|
|-
|-
|44
| [[31/8]]
|2610.651
| +25.809
|[[9/2]]
| +43.543
|
|
|
|
|
|
|-
|-
|45
| [[25/23]]
|2669.984
| -25.811
|[[14/3]]
| -43.547
|
|
|
|
|
|
|-
|-
|46
| [[29/13]]
|2729.317
| -25.814
|[[29/6]]
| -43.553
|
|- style="background-color: #cccccc;"
|
| ''[[25/12]]''
|
| ''-25.979''
|
| ''-43.831''
|
|- style="background-color: #cccccc;"
|
| ''[[9/8]]''
| ''-26.097''
| ''-44.029''
|-
|-
|47
| [[22/7]]
|2788.650
| -26.546
|[[5/1]]
| -44.787
|
|- style="background-color: #cccccc;"
|
| ''[[31/17]]''
|
| ''+26.800''
|
| ''+45.215''
|
|
|-
|-
|51
| [[11/4]]
|3025.982
| +26.815
|[[23/4]]
| +45.242
|
|- style="background-color: #cccccc;"
|
| ''[[33/23]]''
|
| ''+26.984''
|
| ''+45.526''
|
|
|-
|-
|52
| [[23/11]]
|3085.315
| +27.008
|[[6/1]]
| +45.567
|
|- style="background-color: #cccccc;"
|
| ''[[12/11]]''
|
| ''+27.176''
|
| ''+45.851''
|
|
|-
|-
|57
| [[20/11]]
|3381.980
| -27.387
|[[7/1]]
| -46.206
|
|
|
|
|
|
|-
|-
|61
| [[31/7]]
|3619.312
| -27.552
|[[8/1]]
| -46.485
|
|- style="background-color: #cccccc;"
|
| ''[[22/17]]''
|
| ''+27.806''
|
| ''+46.914''
|
|
|-
|-
|63
| [[27/8]]
|3737.978
| +27.895
|[[26/3]]
| +47.063
|
|
|
|
|
|
|-
|-
|64
| [[29/24]]
|3797.311
| +28.004
|[[9/1]]
| +47.248
|
|
|
|
|
|
|-
|-
|67
| [[25/4]]
|3975.310
| +28.012
|[[10/1]]
| +47.261
|
|- style="background-color: #cccccc;"
|
| ''[[32/17]]''
|
| ''-28.165''
|
| ''-47.518''
|
|- style="background-color: #cccccc;"
|
| ''[[31/24]]''
| ''-28.183''
| ''-47.549''
|-
|-
|70
| [[5/4]]
|4153.309
| +28.584
|[[11/1]]
| +48.226
|
|- style="background-color: #cccccc;"
|
| ''[[29/7]]''
|
| ''+28.635''
|
| ''+48.312''
|
|- style="background-color: #cccccc;"
|
| ''[[23/15]]''
| ''-28.752''
| ''-48.510''
|- style="background-color: #cccccc;"
| ''[[27/17]]''
| ''+28.886''
| ''+48.735''
|-
|-
|75
| [[19/4]]
|4449.974
| +28.958
|[[13/1]]
| +48.857
|
|
|
|
|
|
|-
|-
|77
| [[17/2]]
|4568.640
| +29.124
|[[14/1]]
| +49.137
|
|
|
|
|
|
|-
|-
|78
| [[4/1]]
|4627.972
| -29.156
|[[29/2]]
| -49.191
|
|- style="background-color: #cccccc;"
|
| ''[[30/7]]''
|
| ''+29.215''
|
| ''+49.290''
|
|- style="background-color: #cccccc;"
|
| ''[[23/3]]''
| ''-29.324''
| ''-49.475''
|- style="background-color: #cccccc;"
| ''[[18/13]]''
| ''+29.329''
| ''+49.482''
|-
|-
|79
| [[7/6]]
|4687.305
| +29.485
|[[15/1]]
| +49.745
|
|
|
|
|
|
|-
|-
|80
| [[20/19]]
|4746.638
| -29.530
|[[31/2]]
| -49.821
|
|- style="background-color: #cccccc;"
|
| ''[[20/1]]''
|
| ''+29.544''
|
| ''+49.845''
|
|- style="background-color: #cccccc;"
|
| ''[[17/10]]''
| ''-29.575''
| ''-49.898''
|-
|-
|81
| [[23/14]]
|4805.971
| +29.618
|[[16/1]]
| +49.971
|
|
|
|
|
|
|}
|}


== Approximation to JI ==
{| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"
 
|+ style="white-space: nowrap;" | 34-integer-limit intervals in 71zpi with prime 2 removed (by patent val mapping)
{{todo|Apply bold formatting to rows corresponding to prime harmonics and italicize rows that contain inconsistent ratios.|inline=1}}
 
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space:nowrap" |Intervals by direct approximation (even if inconsistent)
|-
|-
! Ratio
! Ratio
Line 622: Line 5,604:
! Error (rel, [[Relative cent|%]])
! Error (rel, [[Relative cent|%]])
|-
|-
|[[14/1]]
| [[34/1]]
|0.186
| -0.032
|0.314
| -0.053
|-
| [[34/19]]
| +0.166
| +0.281
|-
| '''[[19/1]]'''
| '''-0.198'''
| '''-0.334'''
|-
| [[14/3]]
| +0.329
| +0.555
|-
| [[19/5]]
| +0.374
| +0.631
|-
| [[21/13]]
| -0.458
| -0.772
|-
| [[34/5]]
| +0.540
| +0.911
|-
| '''[[5/1]]'''
| '''-0.572'''
| '''-0.965'''
|-
| [[30/29]]
| +0.580
| +0.978
|-
| [[27/10]]
| -0.689
| -1.163
|-
| [[26/9]]
| +0.787
| +1.327
|-
| [[15/14]]
| -0.901
| -1.519
|-
| [[25/19]]
| -0.946
| -1.595
|-
| [[31/22]]
| -1.007
| -1.698
|-
| [[27/22]]
| +1.080
| +1.821
|-
| [[34/25]]
| +1.112
| +1.876
|-
| [[25/1]]
| -1.144
| -1.929
|-
| [[29/6]]
| -1.151
| -1.943
|-
| [[25/11]]
| +1.197
| +2.020
|-
| [[27/2]]
| -1.261
| -2.128
|-
| [[29/28]]
| -1.480
| -2.497
|-
| [[31/28]]
| +1.604
| +2.706
|-
| [[11/5]]
| -1.769
| -2.984
|-
| [[31/6]]
| +1.932
| +3.260
|-
| [[31/27]]
| -2.086
| -3.520
|-
| [[19/11]]
| +2.143
| +3.615
|-
| [[33/26]]
| +2.152
| +3.632
|-
| [[34/11]]
| +2.309
| +3.896
|-
| '''[[11/1]]'''
| '''-2.341'''
| '''-3.949'''
|-
| [[31/30]]
| +2.504
| +4.225
|-
| [[14/11]]
| -2.610
| -4.404
|-
| [[33/14]]
| -2.669
| -4.504
|-
| [[31/10]]
| -2.775
| -4.683
|-
| [[11/3]]
| +2.939
| +4.959
|-
| [[31/29]]
| +3.084
| +5.203
|-
| [[31/2]]
| -3.347
| -5.647
|-
| [[15/11]]
| -3.511
| -5.923
|-
| [[28/27]]
| -3.690
| -6.225
|-
| [[25/14]]
| +3.807
| +6.423
|-
| [[26/15]]
| -3.921
| -6.616
|-
| [[9/2]]
| +4.019
| +6.780
|-
| [[29/22]]
| -4.090
| -6.901
|-
| [[29/18]]
| +4.128
| +6.965
|-
| [[25/3]]
| +4.136
| +6.978
|-
| [[21/1]]
| +4.347
| +7.335
|-
| [[34/21]]
| -4.379
| -7.388
|-
| [[14/5]]
| -4.379
| -7.388
|-
| [[26/3]]
| -4.493
| -7.581
|-
| [[21/19]]
| +4.545
| +7.669
|-
| [[10/9]]
| -4.590
| -7.745
|-
| [[5/3]]
| +4.708
| +7.943
|-
| [[19/14]]
| +4.753
| +8.019
|-
| '''[[13/1]]'''
| '''+4.805'''
| '''+8.107'''
|-
| [[13/7]]
| -4.822
| -8.135
|-
| [[34/13]]
| -4.837
| -8.160
|-
| [[21/5]]
| +4.919
| +8.300
|-
| [[17/7]]
| +4.919
| +8.300
|-
| [[14/1]]
| -4.951
| -8.353
|-
| [[19/13]]
| -5.003
| -8.441
|-
| [[19/3]]
| +5.082
| +8.574
|-
| [[29/27]]
| -5.170
| -8.723
|-
| [[34/3]]
| +5.248
| +8.854
|-
| '''[[3/1]]'''
| '''-5.280'''
| '''-8.908'''
|-
| [[13/5]]
| +5.377
| +9.072
|-
| [[25/21]]
| -5.491
| -9.264
|-
| [[14/9]]
| +5.608
| +9.462
|-
| [[19/15]]
| +5.653
| +9.538
|-
| [[34/15]]
| +5.820
| +9.819
|-
| [[15/1]]
| -5.851
| -9.872
|-
| [[29/10]]
| -5.859
| -9.885
|-
| [[25/13]]
| -5.949
| -10.037
|-
| [[27/26]]
| -6.066
| -10.235
|-
| [[22/9]]
| -6.359
| -10.729
|-
| [[29/2]]
| -6.431
| -10.850
|-
| [[33/25]]
| -6.477
| -10.927
|-
| [[21/11]]
| +6.688
| +11.284
|-
| [[33/2]]
| +6.958
| +11.739
|-
| [[33/5]]
| -7.048
| -11.892
|-
| [[13/11]]
| +7.146
| +12.056
|-
| [[31/18]]
| +7.212
| +12.168
|-
| [[31/9]]
| -7.366
| -12.427
|-
| [[33/19]]
| -7.422
| -12.523
|-
| [[26/11]]
| -7.432
| -12.539
|-
| [[33/10]]
| +7.529
| +12.703
|-
| [[34/33]]
| +7.589
| +12.803
|-
| [[33/1]]
| -7.620
| -12.857
|-
| [[29/4]]
| +8.147
| +13.745
|-
| [[31/26]]
| -8.152
| -13.754
|-
| [[11/9]]
| +8.219
| +13.866
|-
| [[26/25]]
| -8.629
| -14.559
|-
| [[29/20]]
| +8.719
| +14.710
|-
| [[15/2]]
| +8.726
| +14.723
|-
| [[28/9]]
| -8.969
| -15.133
|-
| [[26/5]]
| -9.201
| -15.523
|-
| [[3/2]]
| +9.298
| +15.688
|-
| [[25/9]]
| +9.416
| +15.886
|-
| [[26/19]]
| -9.575
| -16.154
|-
| '''[[7/1]]'''
| '''+9.627'''
| '''+16.242'''
|-
| [[34/7]]
| -9.659
| -16.296
|-
| [[17/13]]
| +9.741
| +16.435
|-
| [[14/13]]
| -9.756
| -16.460
|-
| [[26/1]]
| -9.773
| -16.488
|-
| [[19/7]]
| -9.825
| -16.576
|-
| [[10/3]]
| -9.870
| -16.652
|-
| [[9/5]]
| -9.988
| -16.851
|-
| [[13/3]]
| +10.085
| +17.015
|-
| [[23/17]]
| +10.121
| +17.076
|-
| [[7/5]]
| +10.199
| +17.207
|-
| [[21/17]]
| -10.199
| -17.207
|-
| [[33/31]]
| +10.305
| +17.386
|-
| [[19/9]]
| +10.361
| +17.481
|-
| [[29/9]]
| -10.450
| -17.630
|-
| [[34/9]]
| +10.528
| +17.762
|-
| [[9/1]]
| -10.559
| -17.815
|-
| [[15/13]]
| -10.657
| -17.979
|-
| [[25/7]]
| -10.771
| -18.172
|-
| [[27/14]]
| -10.888
| -18.370
|-
| [[22/15]]
| -11.067
| -18.672
|-
| [[31/4]]
| +11.231
| +18.948
|-
| [[29/26]]
| -11.236
| -18.957
|-
| [[22/3]]
| -11.639
| -19.637
|-
| [[31/20]]
| +11.803
| +19.913
|-
| [[33/28]]
| +11.908
| +20.092
|-
| [[11/7]]
| -11.968
| -20.191
|-
| [[31/15]]
| -12.074
| -20.370
|-
| [[11/2]]
| +12.237
| +20.646
|-
| [[33/13]]
| -12.425
| -20.964
|-
| [[31/3]]
| -12.645
| -21.335
|-
| [[11/10]]
| +12.809
| +21.611
|-
| [[31/14]]
| -12.974
| -21.890
|-
| [[27/4]]
| +13.317
| +22.468
|-
| [[33/29]]
| +13.389
| +22.589
|-
| [[29/12]]
| +13.427
| +22.653
|-
| [[25/2]]
| +13.434
| +22.666
|-
| [[27/11]]
| -13.498
| -22.774
|-
| [[28/15]]
| -13.677
| -23.076
|-
| [[27/20]]
| +13.889
| +23.432
|-
| [[5/2]]
| +14.006
| +23.631
|-
| [[26/21]]
| -14.120
| -23.823
|-
| [[28/3]]
| -14.249
| -24.041
|-
| [[19/2]]
| +14.380
| +24.261
|-
| '''[[17/1]]'''
| '''+14.546'''
| '''+24.542'''
|-
| '''[[2/1]]'''
| '''-14.578'''
| '''-24.595'''
|-
| [[27/25]]
| -14.695
| -24.793
|-
| [[19/17]]
| -14.744
| -24.876
|-
| [[7/3]]
| +14.907
| +25.150
|-
| [[19/10]]
| +14.952
| +25.226
|-
| [[23/7]]
| +15.040
| +25.375
|-
| [[17/5]]
| +15.118
| +25.507
|-
| [[10/1]]
| -15.150
| -25.560
|-
| [[29/15]]
| -15.157
| -25.573
|-
| [[27/5]]
| -15.267
| -25.758
|-
| [[13/9]]
| +15.364
| +25.922
|-
| [[15/7]]
| -15.478
| -26.115
|-
| [[31/11]]
| -15.584
| -26.294
|-
| [[27/19]]
| -15.641
| -26.389
|-
| [[25/17]]
| -15.690
| -26.471
|-
| [[29/3]]
| -15.729
| -26.538
|-
| [[25/22]]
| +15.775
| +26.615
|-
| [[34/27]]
| +15.807
| +26.670
|-
| [[27/1]]
| -15.839
| -26.723
|-
| [[29/14]]
| -16.058
| -27.093
|-
| [[22/5]]
| -16.347
| -27.580
|-
| [[31/12]]
| +16.510
| +27.856
|-
| [[22/19]]
| -16.721
| -28.210
|-
| [[31/25]]
| -16.782
| -28.313
|-
| [[17/11]]
| +16.887
| +28.491
|-
| [[22/1]]
| -16.918
| -28.544
|-
| [[28/11]]
| -17.188
| -28.999
|-
| [[33/7]]
| -17.247
| -29.099
|-
| [[31/5]]
| -17.353
| -29.278
|-
| [[11/6]]
| +17.517
| +29.554
|-
| [[31/19]]
| -17.727
| -29.908
|-
| [[34/31]]
| +17.893
| +30.189
|-
| '''[[31/1]]'''
| '''-17.925'''
| '''-30.243'''
|-
| [[30/11]]
| -18.089
| -30.519
|-
| [[28/25]]
| -18.385
| -31.019
|-
| [[9/4]]
| +18.597
| +31.375
|-
| [[29/11]]
| -18.668
| -31.496
|-
| [[25/6]]
| +18.714
| +31.574
|-
| [[21/2]]
| +18.925
| +31.930
|-
| [[28/5]]
| -18.957
| -31.983
|-
| [[20/9]]
| -19.168
| -32.340
|-
|-
|[[11/5]]
| [[6/5]]
|0.346
| -19.286
|0.583
| -32.538
|-
|-
|[[17/8]]
| [[28/19]]
|0.370
| -19.331
|0.624
| -32.614
|-
|-
|[[31/22]]
| [[13/2]]
|0.388
| +19.383
|0.654
| +32.702
|-
|-
|[[21/13]]
| [[26/7]]
|0.408
| -19.400
|0.688
| -32.731
|-
|-
|[[25/19]]
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|}


{{Stub}}
[[Category:Zeta peak indexes]]
[[Category:Zeta peak indexes]]
Retrieved from "https://en.xen.wiki/w/71zpi"