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'''34 zeta peak index''' (abbreviated '''34zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 34th [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].
'''34 zeta peak index''' (abbreviated '''34zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 34th [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].


{|class="wikitable"
{{ZPI
!colspan="3"|Tuning
| zpi = 34
!colspan="3"|Strength
| steps = 12.0231830072926
!colspan="2"|Closest EDO
| step size = 99.8071807833375
!colspan="2"|Integer limit
| height = 5.193290
|-
| integral = 1.269599
!ZPI
| gap = 15.899282
!Steps per octave
| edo = 12edo
!Step size (cents)
| octave = 1197.68616940005
!Height
| consistent = 10
!Integral
| distinct = 6
!Gap
}}
!EDO
!Octave (cents)
!Consistent
!Distinct
|-
|34zpi
|12.0231830072926
|99.8071807833375
|5.193290
|1.269599
|15.899282
|[[12edo]]
|1197.68616940005
|10
|6
|}


== Intervals ==
== Intervals ==
{| class="wikitable center-1 right-2 left-3 center-4"
{| class="wikitable center-1 right-2 left-3 center-4"
|+ style="font-size: 105%; white-space: nowrap;" | Intervals in 34zpi
|-
|-
| colspan="3" style="text-align:left;" | JI ratios are comprised of 32-integer limit ratios,<br>and are stylized as follows to indicate their accuracy:
| colspan="3" style="text-align:left;" | JI ratios are comprised of 16-integer-limit ratios,<br>and are stylized as follows to indicate their accuracy:
* '''<u>Bold Underlined:</u>''' relative error < 8.333 %
* '''<u>Bold Underlined:</u>''' relative error < 8.333 %
* '''Bold:''' relative error < 16.667 %
* '''Bold:''' relative error < 16.667 %
Line 41: Line 25:
* <small><small>Small Small:</small></small> relative error < 41.667 %
* <small><small>Small Small:</small></small> relative error < 41.667 %
* <small><small><small>Small Small Small:</small></small></small> relative error < 50 %
* <small><small><small>Small Small Small:</small></small></small> relative error < 50 %
| style="text-align:right;" | <center>'''12edo'''</center><br>[[9/8|Whole tone]] = 2 steps<br>[[256/243|Limma]] = 1 step<br>[[2187/2048|Apotome]] = 1 step
| style="text-align:right;" | <center>'''⟨12 19]'''</center><br>[[9/8|Whole tone]] = 2 steps<br>[[256/243|Limma]] = 1 step<br>[[2187/2048|Apotome]] = 1 step
|-
|-
! Degree
! Degree
! Cents
! Cents
! Ratios
! Ratios
! Ups and Downs Notation
! Ups and downs notation
|-
|-
| 0
| 0
Line 55: Line 39:
| 1
| 1
| 99.807
| 99.807
| <small><small><small>[[32/31]]</small></small></small>, <small><small><small>[[31/30]]</small></small></small>, <small><small>[[30/29]]</small></small>, <small><small>[[29/28]]</small></small>, <small><small>[[28/27]]</small></small>, <small><small>[[27/26]]</small></small>, <small>[[26/25]]</small>, <small>[[25/24]]</small>, <small>[[24/23]]</small>, [[23/22]], [[22/21]], '''[[21/20]]''', '''[[20/19]]''', '''<u>[[19/18]]'''</u>, '''<u>[[18/17]]'''</u>, '''<u>[[17/16]]'''</u>, '''[[16/15]]''', '''[[31/29]]''', [[15/14]], [[29/27]], <small>[[14/13]]</small>, <small><small>[[27/25]]</small></small>, <small><small>[[13/12]]</small></small>, <small><small><small>[[25/23]]</small></small></small>
| '''[[16/15]]''', [[15/14]], <small>[[14/13]]</small>, <small><small>[[13/12]]</small></small>
| m2
| m2
|-
|-
| 2
| 2
| 199.614
| 199.614
| <small><small><small>[[12/11]]</small></small></small>, <small><small><small>[[23/21]]</small></small></small>, <small><small>[[11/10]]</small></small>, <small>[[32/29]]</small>, <small>[[21/19]]</small>, [[31/28]], [[10/9]], '''[[29/26]]''', '''<u>[[19/17]]'''</u>, '''<u>[[28/25]]'''</u>, '''<u>[[9/8]]'''</u>, '''[[26/23]]''', [[17/15]], [[25/22]], <small>[[8/7]]</small>, <small><small>[[31/27]]</small></small>, <small><small><small>[[23/20]]</small></small></small>, <small><small><small>[[15/13]]</small></small></small>
| <small><small><small>[[12/11]]</small></small></small>, <small><small>[[11/10]]</small></small>, [[10/9]], '''<u>[[9/8]]'''</u>, <small>[[8/7]]</small>, <small><small><small>[[15/13]]</small></small></small>
| M2
| M2
|-
|-
| 3
| 3
| 299.422
| 299.422
| <small><small><small>[[22/19]]</small></small></small>, <small><small><small>[[29/25]]</small></small></small>, <small>[[7/6]]</small>, [[27/23]], [[20/17]], '''[[13/11]]''', '''<u>[[32/27]]'''</u>, '''<u>[[19/16]]'''</u>, '''<u>[[25/21]]'''</u>, '''<u>[[31/26]]'''</u>, '''[[6/5]]''', <small>[[29/24]]</small>, <small>[[23/19]]</small>, <small><small>[[17/14]]</small></small>, <small><small>[[28/23]]</small></small>, <small><small><small>[[11/9]]</small></small></small>
| <small>[[7/6]]</small>, '''[[13/11]]''', '''[[6/5]]''', <small><small><small>[[11/9]]</small></small></small>
| m3
| m3
|-
|-
| 4
| 4
| 399.229
| 399.229
| <small><small><small>[[27/22]]</small></small></small>, <small><small>[[16/13]]</small></small>, <small><small>[[21/17]]</small></small>, <small>[[26/21]]</small>, <small>[[31/25]]</small>, '''[[5/4]]''', '''<u>[[29/23]]'''</u>, '''<u>[[24/19]]'''</u>, '''[[19/15]]''', [[14/11]], <small>[[23/18]]</small>, <small>[[32/25]]</small>, <small><small>[[9/7]]</small></small>, <small><small><small>[[31/24]]</small></small></small>, <small><small><small>[[22/17]]</small></small></small>
| <small><small>[[16/13]]</small></small>, '''[[5/4]]''', [[14/11]], <small><small>[[9/7]]</small></small>
| M3
| M3
|-
|-
| 5
| 5
| 499.036
| 499.036
| <small><small><small>[[13/10]]</small></small></small>, <small><small>[[30/23]]</small></small>, <small><small>[[17/13]]</small></small>, <small>[[21/16]]</small>, [[25/19]], [[29/22]], '''<u>[[4/3]]'''</u>, [[31/23]], [[27/20]], [[23/17]], <small>[[19/14]]</small>, <small><small>[[15/11]]</small></small>, <small><small><small>[[26/19]]</small></small></small>
| <small><small><small>[[13/10]]</small></small></small>, '''<u>[[4/3]]'''</u>, <small><small>[[15/11]]</small></small>
| P4
| P4
|-
|-
| 6
| 6
| 598.843
| 598.843
| <small><small><small>[[11/8]]</small></small></small>, <small><small>[[29/21]]</small></small>, <small><small>[[18/13]]</small></small>, <small>[[25/18]]</small>, <small>[[32/23]]</small>, '''[[7/5]]''', '''<u>[[31/22]]'''</u>, '''<u>[[24/17]]'''</u>, '''<u>[[17/12]]'''</u>, '''[[27/19]]''', [[10/7]], <small>[[23/16]]</small>, <small><small>[[13/9]]</small></small>, <small><small><small>[[29/20]]</small></small></small>, <small><small><small>[[16/11]]</small></small></small>
| <small><small><small>[[11/8]]</small></small></small>, '''[[7/5]]''', [[10/7]], <small><small>[[13/9]]</small></small>, <small><small><small>[[16/11]]</small></small></small>
| A4, d5
| A4, d5
|-
|-
| 7
| 7
| 698.650
| 698.650
| <small><small><small>[[19/13]]</small></small></small>, <small><small>[[22/15]]</small></small>, <small>[[25/17]]</small>, <small>[[28/19]]</small>, [[31/21]], '''<u>[[3/2]]'''</u>, <small>[[32/21]]</small>, <small><small>[[29/19]]</small></small>, <small><small>[[26/17]]</small></small>, <small><small>[[23/15]]</small></small>, <small><small><small>[[20/13]]</small></small></small>
| '''<u>[[3/2]]'''</u>
| P5
| P5
|-
|-
| 8
| 8
| 798.457
| 798.457
| <small><small><small>[[17/11]]</small></small></small>, <small><small>[[31/20]]</small></small>, <small><small>[[14/9]]</small></small>, <small>[[25/16]]</small>, '''[[11/7]]''', '''<u>[[30/19]]'''</u>, '''<u>[[19/12]]'''</u>, '''<u>[[27/17]]'''</u>, '''[[8/5]]''', <small>[[29/18]]</small>, <small>[[21/13]]</small>, <small><small><small>[[13/8]]</small></small></small>, <small><small><small>[[31/19]]</small></small></small>
| <small><small>[[14/9]]</small></small>, '''[[11/7]]''', '''[[8/5]]''', <small><small><small>[[13/8]]</small></small></small>
| m6
| m6
|-
|-
| 9
| 9
| 898.265
| 898.265
| <small><small><small>[[18/11]]</small></small></small>, <small><small>[[23/14]]</small></small>, <small><small>[[28/17]]</small></small>, '''[[5/3]]''', '''<u>[[32/19]]'''</u>, '''<u>[[27/16]]'''</u>, '''[[22/13]]''', [[17/10]], <small>[[29/17]]</small>, <small><small>[[12/7]]</small></small>, <small><small><small>[[31/18]]</small></small></small>, <small><small><small>[[19/11]]</small></small></small>
| '''[[5/3]]''', <small><small>[[12/7]]</small></small>
| M6
| M6
|-
|-
| 10
| 10
| 998.072
| 998.072
| <small><small><small>[[26/15]]</small></small></small>, <small>[[7/4]]</small>, '''[[30/17]]''', '''[[23/13]]''', '''<u>[[16/9]]'''</u>, '''<u>[[25/14]]'''</u>, [[9/5]], <small>[[29/16]]</small>, <small><small>[[20/11]]</small></small>, <small><small><small>[[31/17]]</small></small></small>
| <small>[[7/4]]</small>, '''<u>[[16/9]]'''</u>, [[9/5]]
| m7
| m7
|-
|-
| 11
| 11
| 1097.879
| 1097.879
| <small><small><small>[[11/6]]</small></small></small>, <small><small>[[24/13]]</small></small>, <small>[[13/7]]</small>, [[28/15]], '''[[15/8]]''', '''<u>[[32/17]]'''</u>, '''<u>[[17/9]]'''</u>, '''[[19/10]]''', [[21/11]], <small>[[23/12]]</small>, <small><small>[[25/13]]</small></small>, <small><small>[[27/14]]</small></small>, <small><small><small>[[29/15]]</small></small></small>, <small><small><small>[[31/16]]</small></small></small>
| <small><small><small>[[11/6]]</small></small></small>, <small>[[13/7]]</small>, '''[[15/8]]'''
| M7
| M7
|-
|-
Line 115: Line 99:
| 13
| 13
| 1297.493
| 1297.493
| <small><small>[[31/15]]</small></small>, <small><small>[[29/14]]</small></small>, <small>[[27/13]]</small>, <small>[[25/12]]</small>, [[23/11]], '''[[21/10]]''', '''<u>[[19/9]]'''</u>, '''<u>[[17/8]]'''</u>, '''[[32/15]]''', [[15/7]], <small>[[28/13]]</small>, <small><small>[[13/6]]</small></small>
| [[15/7]], <small><small>[[13/6]]</small></small>
| m2 +1 oct
| m2 +1 oct
|-
|-
| 14
| 14
| 1397.301
| 1397.301
| <small><small><small>[[24/11]]</small></small></small>, <small>[[11/5]]</small>, [[31/14]], '''[[20/9]]''', '''<u>[[29/13]]'''</u>, '''<u>[[9/4]]'''</u>, [[25/11]], <small><small>[[16/7]]</small></small>, <small><small><small>[[23/10]]</small></small></small>
| <small>[[11/5]]</small>, '''<u>[[9/4]]'''</u>, <small><small>[[16/7]]</small></small>
| M2 +1 oct
| M2 +1 oct
|-
|-
| 15
| 15
| 1497.108
| 1497.108
| <small><small><small>[[30/13]]</small></small></small>, <small>[[7/3]]</small>, '''<u>[[26/11]]'''</u>, '''<u>[[19/8]]'''</u>, '''<u>[[31/13]]'''</u>, [[12/5]], <small>[[29/12]]</small>, <small><small>[[17/7]]</small></small>
| <small>[[7/3]]</small>, [[12/5]]
| m3 +1 oct
| m3 +1 oct
|-
|-
| 16
| 16
| 1596.915
| 1596.915
| <small><small><small>[[22/9]]</small></small></small>, <small><small><small>[[27/11]]</small></small></small>, <small><small>[[32/13]]</small></small>, '''[[5/2]]''', [[28/11]], <small>[[23/9]]</small>, <small><small>[[18/7]]</small></small>, <small><small><small>[[31/12]]</small></small></small>
| '''[[5/2]]'''
| M3 +1 oct
| M3 +1 oct
|-
|-
| 17
| 17
| 1696.722
| 1696.722
| <small><small><small>[[13/5]]</small></small></small>, <small>[[21/8]]</small>, [[29/11]], '''<u>[[8/3]]'''</u>, [[27/10]], <small>[[19/7]]</small>, <small><small>[[30/11]]</small></small>
| <small><small><small>[[13/5]]</small></small></small>, '''<u>[[8/3]]'''</u>
| P4 +1 oct
| P4 +1 oct
|-
|-
| 18
| 18
| 1796.529
| 1796.529
| <small><small><small>[[11/4]]</small></small></small>, <small>[[25/9]]</small>, '''[[14/5]]''', '''<u>[[31/11]]'''</u>, '''<u>[[17/6]]'''</u>, [[20/7]], <small>[[23/8]]</small>, <small><small>[[26/9]]</small></small>, <small><small><small>[[29/10]]</small></small></small>
| <small><small><small>[[11/4]]</small></small></small>, '''[[14/5]]'''
| A4 +1 oct, d5 +1 oct
| A4 +1 oct, d5 +1 oct
|-
|-
| 19
| 19
| 1896.336
| 1896.336
| <small><small><small>[[32/11]]</small></small></small>, '''<u>[[3/1]]'''</u>
| '''<u>[[3/1]]'''</u>
| P5 +1 oct
| P5 +1 oct
|-
|-
| 20
| 20
| 1996.144
| 1996.144
| <small><small>[[31/10]]</small></small>, <small>[[28/9]]</small>, [[25/8]], '''[[22/7]]''', '''<u>[[19/6]]'''</u>, [[16/5]], <small>[[29/9]]</small>, <small><small><small>[[13/4]]</small></small></small>
| [[16/5]], <small><small><small>[[13/4]]</small></small></small>
| m6 +1 oct
| m6 +1 oct
|-
|-
| 21
| 21
| 2095.951
| 2095.951
| <small><small>[[23/7]]</small></small>, '''[[10/3]]''', '''[[27/8]]''', [[17/5]], <small><small>[[24/7]]</small></small>, <small><small><small>[[31/9]]</small></small></small>
| '''[[10/3]]'''
| M6 +1 oct
| M6 +1 oct
|-
|-
| 22
| 22
| 2195.758
| 2195.758
| <small>[[7/2]]</small>, '''<u>[[32/9]]'''</u>, '''<u>[[25/7]]'''</u>, [[18/5]], <small><small>[[29/8]]</small></small>
| <small>[[7/2]]</small>
| m7 +1 oct
| m7 +1 oct
|-
|-
| 23
| 23
| 2295.565
| 2295.565
| <small><small><small>[[11/3]]</small></small></small>, [[26/7]], '''<u>[[15/4]]'''</u>, '''[[19/5]]''', <small>[[23/6]]</small>, <small><small>[[27/7]]</small></small>, <small><small><small>[[31/8]]</small></small></small>
| <small><small><small>[[11/3]]</small></small></small>, '''<u>[[15/4]]'''</u>
| M7 +1 oct
| M7 +1 oct
|-
|-
Line 175: Line 159:
| 25
| 25
| 2495.180
| 2495.180
| <small><small>[[29/7]]</small></small>, [[25/6]], '''[[21/5]]''', '''[[17/4]]''', [[30/7]], <small><small><small>[[13/3]]</small></small></small>
| <small><small><small>[[13/3]]</small></small></small>
| m2 +2 oct
| m2 +2 oct
|-
|-
| 26
| 26
| 2594.987
| 2594.987
| <small>[[22/5]]</small>, [[31/7]], '''[[9/2]]''', <small><small>[[32/7]]</small></small>, <small><small><small>[[23/5]]</small></small></small>
| '''[[9/2]]'''
| M2 +2 oct
| M2 +2 oct
|-
|-
| 27
| 27
| 2694.794
| 2694.794
| <small>[[14/3]]</small>, '''<u>[[19/4]]'''</u>, [[24/5]], <small>[[29/6]]</small>
| <small>[[14/3]]</small>
| m3 +2 oct
| m3 +2 oct
|-
|-
| 28
| 28
| 2794.601
| 2794.601
| '''<u>[[5/1]]'''</u>, <small><small><small>[[31/6]]</small></small></small>
| '''<u>[[5/1]]'''</u>
| M3 +2 oct
| M3 +2 oct
|-
|-
| 29
| 29
| 2894.408
| 2894.408
| <small><small>[[26/5]]</small></small>, [[21/4]], '''<u>[[16/3]]'''</u>, <small>[[27/5]]</small>
| '''<u>[[16/3]]'''</u>
| P4 +2 oct
| P4 +2 oct
|-
|-
| 30
| 30
| 2994.215
| 2994.215
| <small><small><small>[[11/2]]</small></small></small>, '''[[28/5]]''', '''[[17/3]]''', <small><small>[[23/4]]</small></small>, <small><small><small>[[29/5]]</small></small></small>
| <small><small><small>[[11/2]]</small></small></small>
| A4 +2 oct, d5 +2 oct
| A4 +2 oct, d5 +2 oct
|-
|-
Line 210: Line 194:
| 32
| 32
| 3193.830
| 3193.830
| <small><small>[[31/5]]</small></small>, [[25/4]], '''<u>[[19/3]]'''</u>, [[32/5]], <small><small><small>[[13/2]]</small></small></small>
| <small><small><small>[[13/2]]</small></small></small>
| m6 +2 oct
| m6 +2 oct
|-
|-
| 33
| 33
| 3293.637
| 3293.637
| '''[[20/3]]''', '''[[27/4]]'''
|  
| M6 +2 oct
| M6 +2 oct
|-
|-
| 34
| 34
| 3393.444
| 3393.444
| [[7/1]], <small><small>[[29/4]]</small></small>
| [[7/1]]
| m7 +2 oct
| m7 +2 oct
|-
|-
| 35
| 35
| 3493.251
| 3493.251
| <small><small><small>[[22/3]]</small></small></small>, '''<u>[[15/2]]'''</u>, <small>[[23/3]]</small>
| '''<u>[[15/2]]'''</u>
| M7 +2 oct
| M7 +2 oct
|-
|-
| 36
| 36
| 3593.059
| 3593.059
| <small><small><small>[[31/4]]</small></small></small>, '''<u>[[8/1]]'''</u>
| '''<u>[[8/1]]'''</u>
| P1 +3 oct
| P1 +3 oct
|-
|-
| 37
| 37
| 3692.866
| 3692.866
| [[25/3]], '''[[17/2]]''', <small><small><small>[[26/3]]</small></small></small>
|  
| m2 +3 oct
| m2 +3 oct
|-
|-
Line 245: Line 229:
| 39
| 39
| 3892.480
| 3892.480
| <small>[[28/3]]</small>, '''<u>[[19/2]]'''</u>, <small><small>[[29/3]]</small></small>
|  
| m3 +3 oct
| m3 +3 oct
|-
|-
Line 255: Line 239:
| 41
| 41
| 4092.094
| 4092.094
| <small><small><small>[[31/3]]</small></small></small>, [[21/2]], '''<u>[[32/3]]'''</u>
|  
| P4 +3 oct
| P4 +3 oct
|-
|-
| 42
| 42
| 4191.902
| 4191.902
| <small><small>[[11/1]]</small></small>, <small><small>[[23/2]]</small></small>
| <small><small>[[11/1]]</small></small>
| A4 +3 oct, d5 +3 oct
| A4 +3 oct, d5 +3 oct
|-
|-
Line 270: Line 254:
| 44
| 44
| 4391.516
| 4391.516
| [[25/2]], <small><small><small>[[13/1]]</small></small></small>
| <small><small><small>[[13/1]]</small></small></small>
| m6 +3 oct
| m6 +3 oct
|-
|-
| 45
| 45
| 4491.323
| 4491.323
| '''[[27/2]]'''
|  
| M6 +3 oct
| M6 +3 oct
|-
|-
| 46
| 46
| 4591.130
| 4591.130
| [[14/1]], <small><small>[[29/2]]</small></small>
| [[14/1]]
| m7 +3 oct
| m7 +3 oct
|-
|-
Line 290: Line 274:
| 48
| 48
| 4790.745
| 4790.745
| <small><small><small>[[31/2]]</small></small></small>, '''[[16/1]]'''
| '''[[16/1]]'''
| P1 +4 oct
| P1 +4 oct
|}
== Approximation to JI ==
=== Interval mappings ===
The following tables show how 16-integer-limit intervals are represented in 34zpi. 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;" | 16-integer-limit intervals in 34zpi (by direct approximation)
|-
|-
| 49
! Ratio
| 4890.552
! Error (abs, [[Cent|¢]])
| '''[[17/1]]'''
! Error (rel, [[Relative cent|%]])
| m2 +4 oct
|-
|-
| 50
| [[4/3]]
| 4990.359
| +0.991
| '''[[18/1]]'''
| +0.993
| M2 +4 oct
|-
|-
| 51
| [[8/3]]
| 5090.166
| -1.323
| '''<u>[[19/1]]'''</u>
| -1.325
| m3 +4 oct
|-
|-
| 52
| [[16/9]]
| 5189.973
| +1.982
| '''<u>[[20/1]]'''</u>
| +1.986
| M3 +4 oct
|-
|-
| 53
| '''[[2/1]]'''
| 5289.781
| '''-2.314'''
| [[21/1]]
| '''-2.318'''
| P4 +4 oct
|-
|-
| 54
| [[15/1]]
| 5389.588
| +2.669
| <small><small>[[22/1]]</small></small>, <small><small>[[23/1]]</small></small>
| +2.674
| A4 +4 oct, d5 +4 oct
|-
| [[3/2]]
| -3.305
| -3.311
|-
| [[16/3]]
| -3.637
| -3.644
|-
| [[9/8]]
| -4.296
| -4.304
|-
|-
| 55
| [[4/1]]
| 5489.395
| -4.628
| '''[[24/1]]'''
| -4.637
| P5 +4 oct
|-
|-
| 56
| [[15/2]]
| 5589.202
| +4.983
| '''[[25/1]]'''
| +4.992
| m6 +4 oct
|-
|-
| 57
| '''[[3/1]]'''
| 5689.009
| '''-5.619'''
| <small><small><small>[[26/1]]</small></small></small>, [[27/1]]
| '''-5.629'''
| M6 +4 oct
|-
|-
| 58
| [[10/1]]
| 5788.816
| +5.974
| [[28/1]], <small><small>[[29/1]]</small></small>
| +5.985
| m7 +4 oct
|-
|-
| 59
| [[9/4]]
| 5888.624
| -6.609
| '''<u>[[30/1]]'''</u>
| -6.622
| M7 +4 oct
|-
|-
| 60
| [[8/1]]
| 5988.431
| -6.941
| <small><small><small>[[31/1]]</small></small></small>, '''[[32/1]]'''
| -6.955
| P1 +5 oct
|-
| [[15/4]]
| +7.296
| +7.311
|-
| [[6/1]]
| -7.932
| -7.948
|-
| '''[[5/1]]'''
| '''+8.287'''
| '''+8.303'''
|-
| [[9/2]]
| -8.923
| -8.941
|-
| [[16/1]]
| -9.255
| -9.273
|-
| [[15/8]]
| +9.610
| +9.629
|- style="background-color: #cccccc;"
| ''[[13/11]]''
| ''+10.212''
| ''+10.232''
|-
| [[12/1]]
| -10.246
| -10.266
|-
| [[5/2]]
| +10.601
| +10.622
|-
| [[9/1]]
| -11.237
| -11.259
|-
| [[10/3]]
| +11.592
| +11.614
|-
| [[16/15]]
| -11.924
| -11.947
|-
| [[5/4]]
| +12.915
| +12.940
|-
| [[5/3]]
| +13.906
| +13.933
|-
| [[14/5]]
| +14.017
| +14.044
|-
| [[8/5]]
| -15.229
| -15.258
|-
| [[11/7]]
| +15.965
| +15.996
|-
| [[6/5]]
| -16.220
| -16.251
|-
| [[7/5]]
| +16.331
| +16.362
|-
| [[10/9]]
| +17.211
| +17.244
|-
| [[16/5]]
| -17.543
| -17.577
|-
| [[14/11]]
| -18.279
| -18.315
|-
| [[12/5]]
| -18.534
| -18.569
|-
| [[10/7]]
| -18.645
| -18.681
|-
| [[9/5]]
| -19.524
| -19.562
|-
| [[15/14]]
| -19.636
| -19.674
|-
| [[15/7]]
| -21.949
| -21.992
|-
| [[14/1]]
| +22.304
| +22.347
|-
| '''[[7/1]]'''
| '''+24.618'''
| '''+24.666'''
|- style="background-color: #cccccc;"
| ''[[13/7]]''
| ''+26.177''
| ''+26.228''
|-
| [[7/2]]
| +26.932
| +26.984
|-
| [[14/3]]
| +27.923
| +27.977
|- style="background-color: #cccccc;"
| ''[[14/13]]''
| ''-28.491''
| ''-28.546''
|-
| [[7/4]]
| +29.246
| +29.302
|-
| [[7/3]]
| +30.237
| +30.295
|-
| [[8/7]]
| -31.560
| -31.621
|-
| [[11/5]]
| +32.296
| +32.359
|-
| [[7/6]]
| +32.551
| +32.614
|-
| [[14/9]]
| +33.542
| +33.606
|-
| [[16/7]]
| -33.874
| -33.939
|-
| [[11/10]]
| +34.610
| +34.677
|-
| [[12/7]]
| -34.864
| -34.932
|-
| [[9/7]]
| -35.855
| -35.925
|-
| [[13/9]]
| -37.775
| -37.848
|-
| [[15/11]]
| -37.915
| -37.988
|-
| [[13/12]]
| -38.765
| -38.840
|-
| [[16/13]]
| +39.756
| +39.833
|-
| '''[[11/1]]'''
| '''+40.584'''
| '''+40.662'''
|-
| [[13/6]]
| -41.079
| -41.159
|-
| [[13/8]]
| -42.070
| -42.151
|- style="background-color: #cccccc;"
| ''[[13/5]]''
| ''+42.508''
| ''+42.590''
|-
| [[11/2]]
| +42.897
| +42.980
|-
| [[13/3]]
| -43.393
| -43.477
|-
| [[13/4]]
| -44.384
| -44.470
|- style="background-color: #cccccc;"
| ''[[13/10]]''
| ''+44.822''
| ''+44.909''
|-
| [[11/4]]
| +45.211
| +45.299
|-
| [[11/3]]
| +46.202
| +46.291
|-
| [[13/2]]
| -46.698
| -46.788
|-
| [[11/8]]
| +47.525
| +47.617
|- style="background-color: #cccccc;"
| ''[[11/9]]''
| ''-47.986''
| ''-48.079''
|- style="background-color: #cccccc;"
| ''[[15/13]]''
| ''-48.127''
| ''-48.220''
|-
| [[11/6]]
| +48.516
| +48.610
|- style="background-color: #cccccc;"
| ''[[12/11]]''
| ''+48.977''
| ''+49.072''
|-
| '''[[13/1]]'''
| '''-49.012'''
| '''-49.106'''
|-
| [[16/11]]
| -49.839
| -49.935
|}
|}
== Approximation to JI ==


{| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"
{| class="wikitable center-1 right-2 right-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Intervals by direct approximation (even if inconsistent)
|+ style="white-space: nowrap;" | 16-integer-limit intervals in 34zpi (by patent val mapping)
|-
|-
! Ratio
! Ratio
Line 364: Line 616:
|-
|-
| [[4/3]]
| [[4/3]]
| -0.991
| +0.991
| -0.993
| +0.993
|-
|-
| [[8/3]]
| [[8/3]]
| +1.323
| -1.323
| +1.325
| -1.325
|-
|-
| [[16/9]]
| [[16/9]]
| -1.982
| +1.982
| -1.986
| +1.986
|-
|-
| '''[[2/1]]'''
| '''[[2/1]]'''
| '''+2.314'''
| '''-2.314'''
| '''+2.318'''
| '''-2.318'''
|-
|-
| [[15/1]]
| [[15/1]]
| -2.669
| +2.669
| -2.674
| +2.674
|-
|-
| [[3/2]]
| [[3/2]]
| +3.305
| -3.305
| +3.311
| -3.311
|-
|-
| [[16/3]]
| [[16/3]]
| +3.637
| -3.637
| +3.644
| -3.644
|-
|-
| [[9/8]]
| [[9/8]]
| +4.296
| -4.296
| +4.304
| -4.304
|-
|-
| [[4/1]]
| [[4/1]]
| +4.628
| -4.628
| +4.637
| -4.637
|-
|-
| [[15/2]]
| [[15/2]]
| -4.983
| +4.983
| -4.992
| +4.992
|-
|-
| '''[[3/1]]'''
| '''[[3/1]]'''
| '''+5.619'''
| '''-5.619'''
| '''+5.629'''
| '''-5.629'''
|-
|-
| [[10/1]]
| [[10/1]]
| -5.974
| +5.974
| -5.985
| +5.985
|-
|-
| [[9/4]]
| [[9/4]]
| +6.609
| -6.609
| +6.622
| -6.622
|-
|-
| [[8/1]]
| [[8/1]]
| +6.941
| -6.941
| +6.955
| -6.955
|-
|-
| [[15/4]]
| [[15/4]]
| -7.296
| +7.296
| -7.311
| +7.311
|-
|-
| [[6/1]]
| [[6/1]]
| +7.932
| -7.932
| +7.948
| -7.948
|-
|-
| '''[[5/1]]'''
| '''[[5/1]]'''
| '''-8.287'''
| '''+8.287'''
| '''-8.303'''
| '''+8.303'''
|-
|-
| [[9/2]]
| [[9/2]]
| +8.923
| -8.923
| +8.941
| -8.941
|-
|-
| [[16/1]]
| [[16/1]]
| +9.255
| -9.255
| +9.273
| -9.273
|-
|-
| [[15/8]]
| [[15/8]]
| -9.610
| +9.610
| -9.629
| +9.629
|-
| ''[[13/11]]''
| ''-10.212''
| ''-10.232''
|-
|-
| [[12/1]]
| [[12/1]]
| +10.246
| -10.246
| +10.266
| -10.266
|-
|-
| [[5/2]]
| [[5/2]]
| -10.601
| +10.601
| -10.622
| +10.622
|-
|-
| [[9/1]]
| [[9/1]]
| +11.237
| -11.237
| +11.259
| -11.259
|-
|-
| [[10/3]]
| [[10/3]]
| -11.592
| +11.592
| -11.614
| +11.614
|-
|-
| [[16/15]]
| [[16/15]]
| +11.924
| -11.924
| +11.947
| -11.947
|-
|-
| [[5/4]]
| [[5/4]]
| -12.915
| +12.915
| -12.940
| +12.940
|-
|-
| [[5/3]]
| [[5/3]]
| -13.906
| +13.906
| -13.933
| +13.933
|-
|-
| [[14/5]]
| [[14/5]]
| -14.017
| +14.017
| -14.044
| +14.044
|-
|-
| [[8/5]]
| [[8/5]]
| +15.229
| -15.229
| +15.258
| -15.258
|-
|-
| [[11/7]]
| [[11/7]]
| -15.965
| +15.965
| -15.996
| +15.996
|-
|-
| [[6/5]]
| [[6/5]]
| +16.220
| -16.220
| +16.251
| -16.251
|-
|-
| [[7/5]]
| [[7/5]]
| -16.331
| +16.331
| -16.362
| +16.362
|-
|-
| [[10/9]]
| [[10/9]]
| -17.211
| +17.211
| -17.244
| +17.244
|-
|-
| [[16/5]]
| [[16/5]]
| +17.543
| -17.543
| +17.577
| -17.577
|-
|-
| [[14/11]]
| [[14/11]]
| +18.279
| -18.279
| +18.315
| -18.315
|-
|-
| [[12/5]]
| [[12/5]]
| +18.534
| -18.534
| +18.569
| -18.569
|-
|-
| [[10/7]]
| [[10/7]]
| +18.645
| -18.645
| +18.681
| -18.681
|-
|-
| [[9/5]]
| [[9/5]]
| +19.524
| -19.524
| +19.562
| -19.562
|-
|-
| [[15/14]]
| [[15/14]]
| +19.636
| -19.636
| +19.674
| -19.674
|-
|-
| [[15/7]]
| [[15/7]]
| +21.949
| -21.949
| +21.992
| -21.992
|-
|-
| [[14/1]]
| [[14/1]]
| -22.304
| +22.304
| -22.347
| +22.347
|-
|-
| '''[[7/1]]'''
| '''[[7/1]]'''
| '''-24.618'''
| '''+24.618'''
| '''-24.666'''
| '''+24.666'''
|-
| ''[[13/7]]''
| ''-26.177''
| ''-26.228''
|-
|-
| [[7/2]]
| [[7/2]]
| -26.932
| +26.932
| -26.984
| +26.984
|-
|-
| [[14/3]]
| [[14/3]]
| -27.923
| +27.923
| -27.977
| +27.977
|-
| ''[[14/13]]''
| ''+28.491''
| ''+28.546''
|-
|-
| [[7/4]]
| [[7/4]]
| -29.246
| +29.246
| -29.302
| +29.302
|-
|-
| [[7/3]]
| [[7/3]]
| -30.237
| +30.237
| -30.295
| +30.295
|-
|-
| [[8/7]]
| [[8/7]]
| +31.560
| -31.560
| +31.621
| -31.621
|-
|-
| [[11/5]]
| [[11/5]]
| -32.296
| +32.296
| -32.359
| +32.359
|-
|-
| [[7/6]]
| [[7/6]]
| -32.551
| +32.551
| -32.614
| +32.614
|-
|-
| [[14/9]]
| [[14/9]]
| -33.542
| +33.542
| -33.606
| +33.606
|-
|-
| [[16/7]]
| [[16/7]]
| +33.874
| -33.874
| +33.939
| -33.939
|-
|-
| [[11/10]]
| [[11/10]]
| -34.610
| +34.610
| -34.677
| +34.677
|-
|-
| [[12/7]]
| [[12/7]]
| +34.864
| -34.864
| +34.932
| -34.932
|-
|-
| [[9/7]]
| [[9/7]]
| +35.855
| -35.855
| +35.925
| -35.925
|-
|-
| [[13/9]]
| [[13/9]]
| +37.775
| -37.775
| +37.848
| -37.848
|-
|-
| [[15/11]]
| [[15/11]]
| +37.915
| -37.915
| +37.988
| -37.988
|-
|-
| [[13/12]]
| [[13/12]]
| +38.765
| -38.765
| +38.840
| -38.840
|-
|-
| [[16/13]]
| [[16/13]]
| -39.756
| +39.756
| -39.833
| +39.833
|-
|-
| '''[[11/1]]'''
| '''[[11/1]]'''
| '''-40.584'''
| '''+40.584'''
| '''-40.662'''
| '''+40.662'''
|-
|-
| [[13/6]]
| [[13/6]]
| +41.079
| -41.079
| +41.159
| -41.159
|-
|-
| [[13/8]]
| [[13/8]]
| +42.070
| -42.070
| +42.151
| -42.151
|-
| ''[[13/5]]''
| ''-42.508''
| ''-42.590''
|-
|-
| [[11/2]]
| [[11/2]]
| -42.897
| +42.897
| -42.980
| +42.980
|-
|-
| [[13/3]]
| [[13/3]]
| +43.393
| -43.393
| +43.477
| -43.477
|-
|-
| [[13/4]]
| [[13/4]]
| +44.384
| -44.384
| +44.470
| -44.470
|-
| ''[[13/10]]''
| ''-44.822''
| ''-44.909''
|-
|-
| [[11/4]]
| [[11/4]]
| -45.211
| +45.211
| -45.299
| +45.299
|-
|-
| [[11/3]]
| [[11/3]]
| -46.202
| +46.202
| -46.291
| +46.291
|-
|-
| [[13/2]]
| [[13/2]]
| +46.698
| -46.698
| +46.788
| -46.788
|-
|-
| [[11/8]]
| [[11/8]]
| -47.525
| +47.525
| -47.617
| +47.617
|-
| ''[[11/9]]''
| ''+47.986''
| ''+48.079''
|-
| ''[[15/13]]''
| ''+48.127''
| ''+48.220''
|-
|-
| [[11/6]]
| [[11/6]]
| -48.516
| +48.516
| -48.610
| +48.610
|-
| ''[[12/11]]''
| ''-48.977''
| ''-49.072''
|-
|-
| '''[[13/1]]'''
| '''[[13/1]]'''
| '''+49.012'''
| '''-49.012'''
| '''+49.106'''
| '''-49.106'''
|-
|-
| [[16/11]]
| [[16/11]]
| +49.839
| -49.839
| +49.935
| -49.935
|- style="background-color: #cccccc;"
| ''[[12/11]]''
| ''-50.830''
| ''-50.928''
|- style="background-color: #cccccc;"
| ''[[15/13]]''
| ''+51.680''
| ''+51.780''
|- style="background-color: #cccccc;"
| ''[[11/9]]''
| ''+51.821''
| ''+51.921''
|- style="background-color: #cccccc;"
| ''[[13/10]]''
| ''-54.985''
| ''-55.091''
|- style="background-color: #cccccc;"
| ''[[13/5]]''
| ''-57.299''
| ''-57.410''
|- style="background-color: #cccccc;"
| ''[[14/13]]''
| ''+71.316''
| ''+71.454''
|- style="background-color: #cccccc;"
| ''[[13/7]]''
| ''-73.630''
| ''-73.772''
|- style="background-color: #cccccc;"
| ''[[13/11]]''
| ''-89.595''
| ''-89.768''
|}
|}


Retrieved from "https://en.xen.wiki/w/34zpi"