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| <span style="display: block; text-align: right;">[[:de:Bohlen-Pierce|Deutsch]]</span>
| | {{interwiki |
| | |en = Bohlen-Pierce |
| | |de = Bohlen-Pierce |
| | }} |
| | {{Wikipedia}} |
| | [[File:vaisvil-BP-guitar-052-crop.jpg|thumb|x00px|Chris Vaisvil's BP electric guitar. [http://chrisvaisvil.com/the-bohlen-pierce-epiphone-roadie-guitar/ Music from this guitar.]]] |
| | [[File:Sword_BP_guitars.jpg|thumb|300px]] |
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| The '''Bohlen-Pierce''' ('''BP''') scale is a [[nonoctave]] scale, a 13-part equal division of the perfect-twelfth ([[3/1]]) or [[Tritave]] ('''13edt'''). Each step is about 146 ¢, making it a [[macrotonal]] scale. It is closely related to the rank two temperament [[Sensamagic clan#Bohpier|bohpier]]. Bohlen-Pierce is normally thought of (if not in these terms, then in fact) as a temperament defined on the 3.5.7 [[Just intonation subgroups|subgroup]]. However, it (or at least 3.5.7-limit [[edt|13edt]]) can be extended to the 3.5.7.11/4 subgroup. This extension is controversial because of the presence of 2 in the denominator of 11/4, but the interval is present in the sense that 3^(12\13) provides an approximation to it. Chords of Bohlen-Pierce, from this extended perspective, may be found listed on the page [[chords of bohpier]]. Bohlen-Pierce was discovered independently by [[Heinz Bohlen]], [[John Pierce]], [[Kees van Prooijen]], and perhaps others, usually noticed for its good approximation of odd-number just ratios 3:5, 5:7, 3:7, etc.; but not necessarily 4:11, 5:6, 6:7, etc. | | The '''Bohlen–Pierce scale''' ('''BP''') is a 13-tone [[macrotonal]] [[nonoctave]] [[scale]] designed to emphasize odd-number intervals and chords, such as the 3:5:7:9 tetrad. It was first described as a 7-limit [[just intonation]] scale and as an [[equal-step tuning|equal temperament]], [[13edt|13 equal divisions of the tritave]]. The [[tritave]] (3/1) usually replaces the [[octave]] in the role of the [[equave]], such that intervals a tritave apart are considered [[equivalent]]. |
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| [[File:vaisvil-BP-guitar-052-crop.jpg|frame|Chris Vaisvil's BP electric guitar. [http://chrisvaisvil.com/the-bohlen-pierce-epiphone-roadie-guitar/ Music from this guitar.]] | | It is closely related to the rank two temperament [[Sensamagic clan #Bohpier|bohpier]]. It is normally thought of (if not in these terms, then in fact) as a temperament defined on the 3.5.7 [[Just intonation subgroup|subgroup]]. However, it can be extended to the 3.5.7.11/4 subgroup, especially when considering 13edt instead of the JI version. This extension is controversial because of the presence of 2 in the denominator of 11/4, but the interval is present in the sense that 3^(12\13) provides an approximation to it. Chords of Bohlen–Pierce, from this extended perspective, may be found listed on the page [[chords of bohpier]]. |
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| [[File:Sword_BP_guitars.jpg|thumb|500px]] | | Bohlen–Pierce was discovered independently by [[Heinz Bohlen]], [[John Pierce]], [[Kees van Prooijen]], and perhaps others, usually noticed for its good approximation of odd-number just ratios 3:5, 5:7, 3:7, etc.; but not necessarily 4:11, 5:6, 6:7, etc. |
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| =The Rank-2 "Lambda" Temperament and the BP nonatonic scale = | | == Theory == |
| | * [[Relationship between Bohlen–Pierce and octave-ful temperaments]] |
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| Suggested for use as a "diatonic scale" when playing Bohlen-Pierce is the 9-note Lambda scale, which is the 4L5s MOS. This can be thought of as an MOS generated by a 3.5.7 rank-2 temperament that eliminates only the comma 245/243, so that 9/7 * 9/7 = 5/3.
| | === Lambda scale === |
| | {{Main|4L 5s (3/1-equivalent)}} |
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| This is a very good temperament on the 3.5.7 subgroup, and additionally is supported by many EDT's (and even EDOs!) besides 13-EDT.
| | == Intervals == |
| | {{Main|Intervals of BP}} |
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| Some low-numbered EDOs that support Lambda are 19, 22, 27, 41, and 46, all of which make it possible to play BP music to some reasonable extent. These EDOs contain not only the Lambda BP diatonic scale, but also the 13-note "Lambda chromatic" MOS scale, or Lambda[13], which can be thought of as a "detempered" version of the 13-EDT Bohlen Pierce scale. This scale may be a suitable melodic substitute for the BP chromatic scale, and is basically the same as how 19-EDO and 31-EDO do not contain 12-EDO as a subset, but they do contain the meantone[12] chromatic scale.
| | == Variations == |
| | === Triple Bohlen–Pierce === |
| | Proposed by [[Paul Erlich]] is [[39edt]], also known as the ''Triple Bohlen–Pierce scale''. It adds to 13edt accurate approximations to the 11th and 13th harmonics ([[11/9]] and [[13/9]]) and can be used in a variety of ways, for both just intonation chords and harmonies, as standard Bohlen–Pierce scale interlocking three times with calm-sounding quarter-tones, and for various JI modulations. |
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| When playing this temperament in some EDO, it may be desired to stretch/compress the tuning so that the tritave is pure, rather than the octave being pure - or in general, to minimize the error on the 3.5.7 subgroup while ignoring the error on 2/1.
| | == Regular temperament properties == |
| | {{Main| 13edt#Regular temperament properties }} |
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| One can "add" the octave to Lambda temperament by simply creating a new mapping for 2/1. A simple way to do so is to map the 2/1 to +7 of the ~9/7 generators, minus a single tritave. This is Sensi temperament, in essence treating it as a "3.5.7.2 extension" of the original 3.5.7 Lambda temperament.
| | == Instruments == |
| | * [http://www.huygens-fokker.org/bpsite/instruments.html#anchor39641 Bohlen Pierce guitar] |
| | * [[Clarinet#Bohlen-Pierce clarinets|Clarinets]] |
| | * Metallophone |
| | * Electronic Organ |
| | * Stredici |
| | * [http://bp.b0b.com/2016/01/tuning-a-kalimba-to-bp/ Kalimba] (Mbira) |
| | * [http://bp.b0b.com/2012/11/sierra-pedal-steel/ Pedal Steel Guitar] |
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| == List of EDT's supporting Lambda Temperament == | | == Music == |
| | {{Main| Bohlen–Pierce scale/Music }} |
| | {{Catrel|Bohlen-Pierce tracks}} |
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| | == See also == |
| | * [[Catalog of 3.5.7 subgroup rank two temperaments]] |
| | * [[Lumatone mapping for Bohlen–Pierce]] |
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| Below is a list of the equal-temperaments which contain a 4L+5s scale using generators between 422.7 cents and 475.5 cents.
| | == Further reading == |
| | | * [[Sword, Ron]]. [https://ronsword.bigcartel.com/product/creative-applications-for-polyscales-and-scale-sequences ''Bohlen-Pierce Scales for Guitar: Includes Paul Erlich's Triple BP Scales. Notation methods, Chord-Scales, Melody Chords, Polyscales, and new Scales for the Bohlen-Pierce tuning'']. 2009. ([http://www.metatonalmusic.com/books.html Metatonal Music link]) |
| L=1 s=0 4 edt
| |
| | |
| L=1 s=1 9 edt (5flat40 7sharp18)
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| | |
| L=2 s=1 13 (5flat7 7flat3)
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| | |
| L=3 s=1 17 (5sharp10 7flat12)
| |
| | |
| L=3 s=2 22 (~14edo)
| |
| | |
| L=4 s=1 21
| |
| | |
| L=4 s=3 31
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| | |
| L=5 s=1 25
| |
| | |
| L=5 s=2 30 (~19edo) (5sharp3 7flat8)
| |
| | |
| L=5 s=3 35 (~22edo) (5flat14 7sharp0)
| |
| | |
| L=5 s=4 40
| |
| | |
| L=6 s=1 29
| |
| | |
| L=6 s=5 49 (~31EDO) (5sharp8 7sharp8) (Schism*)
| |
| | |
| L=7 s=1 33
| |
| | |
| L=7 s=2 38 (~24edo)
| |
| | |
| L=7 s=3 43 (~27edo) (5sharp0 7flat6)
| |
| | |
| L=7 s=4 48 (5flat13 7flat0)
| |
| | |
| L=7 s=5 53
| |
| | |
| L=7 s=6 58 5sharp1 7sharp10 (Schism*)
| |
| | |
| *Schism, by which I mean, the most accurate value for 5/3 and-or 7/3 is found outside the 4L+5s MOS.
| |
| | |
| [Also, the way I see it, as 4edt and 9edt are comparable to 5edo and 7edo, then the "counterparts" of Blackwood and Whitewood would be found in multiples therein and would be octatonic and octadecatonic, e.g. 12edt and 27edt.] | |
| | |
| {| class="wikitable" style="text-align:center;"
| |
| |-
| |
| ! colspan="7" | Generator
| |
| ! | cents
| |
| hekts
| |
| ! | L
| |
| ! | s
| |
| ! | notes
| |
| |-
| |
| | | 1/4
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | colspan="2" | 475.489
| |
| 325
| |
| | | 0
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 8/33
| |
| | | 461.08
| |
| 315.1515
| |
| | | 403.445
| |
| 275.758
| |
| | | 57.635
| |
| 39.394
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 7/29
| |
| | |
| |
| | | 459.093
| |
| 313.793
| |
| | | 393.508
| |
| 268.9655
| |
| | | 65.585
| |
| 44.828
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 13/54
| |
| | | 457.878
| |
| 312.963
| |
| | | 387.435
| |
| 264.815
| |
| | | 70.428
| |
| 48.148
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 6/25
| |
| | |
| |
| | |
| |
| | | 456.469
| |
| 312
| |
| | | 380.391
| |
| 260
| |
| | | 76.078
| |
| 52
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 17/71
| |
| | | 455.398
| |
| 311.267
| |
| | | 375.033
| |
| 256.338
| |
| | | 80.364
| |
| 54.9295
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 11/46
| |
| | |
| |
| | | 454.815
| |
| 310.8695
| |
| | | 372.122
| |
| 254.348
| |
| | | 82.694
| |
| 56.522
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 16/67
| |
| | | 454.198
| |
| 310.448
| |
| | | 369.036
| |
| 252.239
| |
| | | 85.162
| |
| 58.209
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | | 5/21
| |
| | |
| |
| | |
| |
| | |
| |
| | | 452.846
| |
| 309.524
| |
| | | 362.277
| |
| 247.619
| |
| | | 90.569
| |
| 61.905
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 19/80
| |
| | | 451.714
| |
| 308.75
| |
| | | 356.617
| |
| 243.75
| |
| | | 95.098
| |
| 65
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 14/59
| |
| | |
| |
| | | 451.311
| |
| 308.4745
| |
| | | 354.602
| |
| 242.373
| |
| | | 96.71
| |
| 66.102
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 23/97
| |
| | | 450.979
| |
| 308.247
| |
| | | 352.94
| |
| 241.234
| |
| | | 98.039
| |
| 67.01
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 9/38
| |
| | |
| |
| | |
| |
| | | 450.463
| |
| 307.895
| |
| | | 350.36
| |
| 239.474
| |
| | | 100.103
| |
| 68.421
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 22/93
| |
| | | 449.925
| |
| 307.527
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| | | 347.669
| |
| 237.634
| |
| | | 102.256
| |
| 69.8925
| |
| | |
| |
| |-
| |
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| |
| | |
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| | |
| |
| | |
| |
| | |
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| | | 13/55
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| | |
| |
| | | 449.553
| |
| 307.273
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| | | 345.81
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| 236.364
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| | | 103.743
| |
| 70.909
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| | |
| |
| |-
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| |
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| | |
| |
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| |
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| | |
| |
| | | 17/72
| |
| | | 449.073
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| 306.944
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| | | 343.4085
| |
| 234.722
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| | | 105.664
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| 72.222
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 448.421
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| 306.499
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| | | 340.148
| |
| 232.494
| |
| | | 108.2725
| |
| 74.005
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | | 4/17
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 447.518
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| 305.882
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| | | 335.639
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| 229.412
| |
| | | 111.88
| |
| 76.471
| |
| | | Canonical BP scales are between here...
| |
| |-
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| |
| | |
| |
| | |
| |
| | |
| |
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| |
| | |
| |
| | | 19/81
| |
| | | 446.137
| |
| 304.938
| |
| | | 328.733
| |
| 224.691
| |
| | | 117.405
| |
| 80.247
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 15/64
| |
| | |
| |
| | | 445.771
| |
| 304.6875
| |
| | | 326.8985
| |
| 223.4375
| |
| | | 118.872
| |
| 81.25
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 445.533
| |
| 304.525
| |
| | | 325.711
| |
| 222.626
| |
| | | 119.822
| |
| 81.899
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 26/111
| |
| | | 445.503
| |
| 304.5045
| |
| | | 325.559
| |
| 222.5225
| |
| | | 119.943
| |
| 81.982
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 11/47
| |
| | |
| |
| | |
| |
| | | 445.138
| |
| 304.255
| |
| | | 323.737
| |
| 221.277
| |
| | | 121.401
| |
| 82.989
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 29/124
| |
| | | 444.812
| |
| 304.032
| |
| | | 322.105
| |
| 220.161
| |
| | | 122.705
| |
| 83.871
| |
| | | Golden BP is near here
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 18/77
| |
| | |
| |
| | | 444.613
| |
| 303.896
| |
| | | 321.109
| |
| 219.4805
| |
| | | 123.5035
| |
| 84.416
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 25/107
| |
| | | 444.382
| |
| 303.738
| |
| | | 319.955
| |
| 218.692
| |
| | | 120.427
| |
| 85.047
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | | 7/30
| |
| | |
| |
| | |
| |
| | |
| |
| | | 443.7895
| |
| 303.333
| |
| | | 316.9925
| |
| 216.667
| |
| | | 126.797
| |
| 86.667
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 24/103
| |
| | | 443.174
| |
| 302.913
| |
| | | 313.915
| |
| 214.563
| |
| | | 129.26
| |
| 88.3495
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 17/73
| |
| | |
| |
| | | 442.921
| |
| 302.74
| |
| | | 312.65
| |
| 213.699
| |
| | | 130.271
| |
| 89.041
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 27/116
| |
| | | 442.696
| |
| 302.586
| |
| | | 311.527
| |
| 212.931
| |
| | | 131.169
| |
| 89.65
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 10/43
| |
| | |
| |
| | |
| |
| | | 442.315
| |
| 302.326
| |
| | | 309.621
| |
| 211.628
| |
| | | 132.6945
| |
| 90.698
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 23/99
| |
| | | 441.868
| |
| 302.02
| |
| | | 307.387
| |
| 210.101
| |
| | | 134.482
| |
| 91.919
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 13/56
| |
| | |
| |
| | | 441.525
| |
| 301.786
| |
| | | 305.671
| |
| 208.929
| |
| | | 135.854
| |
| 92.857
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 16/69
| |
| | | 441.033
| |
| 301.449
| |
| | | 303.21
| |
| 207.246
| |
| | | 137.823
| |
| 94.203
| |
| | |
| |
| |-
| |
| | |
| |
| | | 3/13
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 438.913
| |
| 300
| |
| | | 292.6085
| |
| 200
| |
| | | 146.304
| |
| 100
| |
| | | ...and here
| |
| | |
| Boundary of propriety for Lambda scale
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 17/74
| |
| | | 436.935
| |
| 298.649
| |
| | | 282.723
| |
| 193.243
| |
| | | 154.2125
| |
| 105.405
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 14/61
| |
| | |
| |
| | | 436.514
| |
| 298.361
| |
| | | 280.616
| |
| 191.803
| |
| | | 155.897
| |
| 106.557
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 25/109
| |
| | | 436.228
| |
| 298.165
| |
| | | 279.186
| |
| 190.826
| |
| | | 157.042
| |
| 207.339
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 11/48
| |
| | |
| |
| | |
| |
| | | 435.865
| |
| 297.917
| |
| | | 277.368
| |
| 189.583
| |
| | | 158.496
| |
| 108.333
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 30/131
| |
| | | 435.562
| |
| 297.71
| |
| | | 275.856
| |
| 188.55
| |
| | | 159.706
| |
| 109.16
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 19/83
| |
| | |
| |
| | | 435.387
| |
| 297.59
| |
| | | 274.981
| |
| 187.952
| |
| | | 160.406
| |
| 109.639
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 27/118
| |
| | | 435.193
| |
| 297.458
| |
| | | 274.0105
| |
| 187.288
| |
| | | 161.183
| |
| 110.1695
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | | 8/35
| |
| | |
| |
| | |
| |
| | |
| |
| | | 434.733
| |
| 297.143
| |
| | | 271.707
| |
| 185.714
| |
| | | 163.025
| |
| 111.429
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 29/127
| |
| | | 434.305
| |
| 296.85
| |
| | | 269.568
| |
| 184.252
| |
| | | 164.736
| |
| 112.598
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 21/92
| |
| | |
| |
| | | 434.142
| |
| 596.739
| |
| | | 268.7545
| |
| 183.696
| |
| | | 165.387
| |
| 113.0435
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 34/149
| |
| | | 434.003
| |
| 296.644
| |
| | | 268.061
| |
| 183.2215
| |
| | | 165.942
| |
| 113.423
| |
| | | Golden Lambda scale is near here
| |
| | |
| 18\7*30\11=7
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 13/57
| |
| | |
| |
| | |
| |
| | | 433.779
| |
| 296.491
| |
| | | 266.941
| |
| 182.456
| |
| | | 166.838
| |
| 114.035
| |
| | | 18\7*30\11=7
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 31/136
| |
| | | 433.534
| |
| 296.3235
| |
| | | 265.714
| |
| 181.618
| |
| | | 167.8195
| |
| 114.706
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 18/79
| |
| | |
| |
| | | 433.356
| |
| 296.2025
| |
| | | 264.829
| |
| 181.013
| |
| | | 168.528
| |
| 115.189
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 23/101
| |
| | | 433.1185
| |
| 296.04
| |
| | | 263.637
| |
| 180.198
| |
| | | 169.484
| |
| 115.842
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | | 5/22
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 432.2625
| |
| 295.4545
| |
| | | 259.3575
| |
| 177.273
| |
| | | 172.905
| |
| 118.182
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 22/97
| |
| | | 431.371
| |
| 294.845
| |
| | | 254.901
| |
| 174.227
| |
| | | 176.47
| |
| 120.619
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 17/75
| |
| | |
| |
| | | 431.11
| |
| 294.667
| |
| | | 253.594
| |
| 173.333
| |
| | | 177.516
| |
| 121.333
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 29/128
| |
| | | 430.912
| |
| 294.531
| |
| | | 252.603
| |
| 172.626
| |
| | | 178.308
| |
| 121.875
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 12/53
| |
| | |
| |
| | |
| |
| | | 430.631
| |
| 294.334
| |
| | | 251.202
| |
| 171.698
| |
| | | 179.43
| |
| 122.6415
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 31/137
| |
| | | 430.369
| |
| 294.161
| |
| | | 249.892
| |
| 170.803
| |
| | | 180.4775
| |
| 123.358
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 19/84
| |
| | |
| |
| | | 430.204
| |
| 294.048
| |
| | | 249.066
| |
| 170.238
| |
| | | 181.139
| |
| 123.8095
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 26/115
| |
| | | 430.007
| |
| 293.913
| |
| | | 248.081
| |
| 169.565
| |
| | | 181.927
| |
| 124.348
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | | 7/31
| |
| | |
| |
| | |
| |
| | |
| |
| | | 429.474
| |
| 293.548
| |
| | | 245.4135
| |
| 167.742
| |
| | | 184.06
| |
| 125.8065
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 23/102
| |
| | | 428.872
| |
| 293.137
| |
| | | 242.406
| |
| 165.686
| |
| | | 186.466
| |
| 127.451
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 16/71
| |
| | |
| |
| | | 428.6095
| |
| 292.958
| |
| | | 241.093
| |
| 164.789
| |
| | | 187.517
| |
| 128.169
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 25/111
| |
| | | 428.368
| |
| 292.793
| |
| | | 239.886
| |
| 163.964
| |
| | | 188.482
| |
| 128.829
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 9/40
| |
| | |
| |
| | |
| |
| | | 427.94
| |
| 292.5
| |
| | | 237.744
| |
| 162.5
| |
| | | 190.1955
| |
| 130
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 20/89
| |
| | | 427.406
| |
| 292.135
| |
| | | 235.073
| |
| 160.674
| |
| | | 192.3325
| |
| 131.461
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 11/49
| |
| | |
| |
| | | 426.9695
| |
| 291.837
| |
| | | 232.892
| |
| 159.184
| |
| | | 194.077
| |
| 132.653
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 13/58
| |
| | | 426.3
| |
| 291.379
| |
| | | 229.546
| |
| 156.897
| |
| | | 196.754
| |
| 134.483
| |
| | |
| |
| |-
| |
| | | 2/9
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 422.657
| |
| 288.889
| |
| | colspan="2" | 211.328
| |
| 144.444
| |
| | | Separatrix of Lambda and Anti-Lambda scales
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 13/59
| |
| | | 419.075
| |
| 286.441
| |
| | | 225.656
| |
| 154.237
| |
| | | 193.419
| |
| 132.203
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 11/50
| |
| | |
| |
| | | 418.43
| |
| 286
| |
| | | 228.235
| |
| 156
| |
| | | 190.1955
| |
| 130
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 20/91
| |
| | | 418.01
| |
| 285.714
| |
| | | 229.907
| |
| 157.143
| |
| | | 188.105
| |
| 128.571
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 9/41
| |
| | |
| |
| | |
| |
| | | 417.502
| |
| 285.365
| |
| | | 231.946
| |
| 158.537
| |
| | | 185.557
| |
| 126.829
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 25/114
| |
| | | 417.095
| |
| 285.088
| |
| | | 233.573
| |
| 159.649
| |
| | | 183.522
| |
| 125.439
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 16/73
| |
| | |
| |
| | | 416.867
| |
| 284.9315
| |
| | | 234.488
| |
| 160.273
| |
| | | 182.379
| |
| 124.6575
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 23/105
| |
| | | 416.619
| |
| 284.762
| |
| | | 235.48
| |
| 160.952
| |
| | | 181.139
| |
| 123.8095
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | | 7/32
| |
| | |
| |
| | |
| |
| | |
| |
| | | 416.053
| |
| 284.375
| |
| | | 237.744
| |
| 162.5
| |
| | | 178.308
| |
| 121.875
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 26/119
| |
| | | 415.553
| |
| 284.034
| |
| | | 239.742
| |
| 163.8655
| |
| | | 175.811
| |
| 120.168
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 19/87
| |
| | |
| |
| | | 415.3695
| |
| 283.908
| |
| | | 240.477
| |
| 164.368
| |
| | | 174.892
| |
| 119.54
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 31/142
| |
| | | 415.2155
| |
| 283.803
| |
| | | 241.093
| |
| 164.789
| |
| | | 174.123
| |
| 119.014
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 12/55
| |
| | |
| |
| | |
| |
| | | 414.972
| |
| 283.636
| |
| | | 242.067
| |
| 165.4545
| |
| | | 172.905
| |
| 118.182
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 29/133
| |
| | | 414.712
| |
| 283.459
| |
| | | 243.107
| |
| 166.165
| |
| | | 171.605
| |
| 117.293
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 17/78
| |
| | |
| |
| | | 414.528
| |
| 283.333
| |
| | | 243.84
| |
| 166.667
| |
| | | 170.688
| |
| 116.667
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 22/101
| |
| | | 414.287
| |
| 283.168
| |
| | | 244.806
| |
| 167.327
| |
| | | 169.481
| |
| 115.842
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | | 5/23
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 413.4685
| |
| 282.609
| |
| | | 248.081
| |
| 169.565
| |
| | | 165.387
| |
| 113.0435
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 23/106
| |
| | | 412.688
| |
| 282.0755
| |
| | | 251.202
| |
| 171.698
| |
| | | 161.487
| |
| 110.377
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 18/83
| |
| | |
| |
| | | 412.472
| |
| 281.928
| |
| | | 252.066
| |
| 172.289
| |
| | | 160.406
| |
| 109.639
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 31/143
| |
| | | 412.312
| |
| 281.818
| |
| | | 252.707
| |
| 172.727
| |
| | | 159.605
| |
| 109.091
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 13/60
| |
| | |
| |
| | |
| |
| | | 412.09
| |
| 281.667
| |
| | | 253.594
| |
| 173.333
| |
| | | 158.496
| |
| 108.333
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 34/157
| |
| | | 411.888
| |
| 281.529
| |
| | | 254.402
| |
| 173.885
| |
| | | 157.487
| |
| 107.643
| |
| | | Golden Anti-Lambda scale is near here
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 21/97
| |
| | |
| |
| | | 411.7635
| |
| 281.443
| |
| | | 254.901
| |
| 174.227
| |
| | | 156.862
| |
| 107.2165
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 29/134
| |
| | | 411.617
| |
| 281.343
| |
| | | 255.4865
| |
| 174.627
| |
| | | 156.131
| |
| 103.716
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | | 8/37
| |
| | |
| |
| | |
| |
| | |
| |
| | | 411.2335
| |
| 281.081.
| |
| | | 257.021
| |
| 175.676
| |
| | | 154.2125
| |
| 105.405
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 27/125
| |
| | | 410.822
| |
| 280.8
| |
| | | 258.666
| |
| 176.8
| |
| | | 152.156
| |
| 104
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 19/88
| |
| | |
| |
| | | 410.649
| |
| 280.682
| |
| | | 259.3575
| |
| 177.273
| |
| | | 151.292
| |
| 103.409
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 30/139
| |
| | | 410.494
| |
| 280.5755
| |
| | | 259.9795
| |
| 177.698
| |
| | | 150.514
| |
| 102.878
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 11/51
| |
| | |
| |
| | |
| |
| | | 410.2255
| |
| 280.392
| |
| | | 261.053
| |
| 178.431
| |
| | | 149.173
| |
| 101.961
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 25/116
| |
| | | 409.904
| |
| 280.172
| |
| | | 262.339
| |
| 179.31
| |
| | | 147.5655
| |
| 100.862
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 14/65
| |
| | |
| |
| | | 409.652
| |
| 280
| |
| | | 263.348
| |
| 180
| |
| | | 146.304
| |
| 100
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 17/79
| |
| | | 409.2815
| |
| 279.747
| |
| | | 264.819
| |
| 181.013
| |
| | | 144.452
| |
| 98.734
| |
| | |
| |
| |-
| |
| | |
| |
| | | 3/14
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 407.562
| |
| 278.571
| |
| | | 271.708
| |
| 185.714
| |
| | | 135.854
| |
| 92.857
| |
| | | Boundary of propriety for Anti-Lambda scale
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 16/75
| |
| | | 405.75
| |
| 277.333
| |
| | | 278.953
| |
| 190.667
| |
| | | 126.797
| |
| 86.667
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 13/61
| |
| | |
| |
| | | 405.335
| |
| 277.049
| |
| | | 280.616
| |
| 191.803
| |
| | | 124.718
| |
| 85.245
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 23/108
| |
| | | 405.046
| |
| 276.852
| |
| | | 281.771
| |
| 192.593
| |
| | | 123.275
| |
| 84.259
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 10/47
| |
| | |
| |
| | |
| |
| | | 404.671
| |
| 276.596
| |
| | | 283.27
| |
| 193.617
| |
| | | 121.401
| |
| 82.979
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 27/127
| |
| | | 404.353
| |
| 276.378
| |
| | | 284.544
| |
| 194.488
| |
| | | 119.808
| |
| 81.89
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 17/80
| |
| | |
| |
| | | 404.165
| |
| 276.25
| |
| | | 285.293
| |
| 195
| |
| | | 118.873
| |
| 81.25
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 24/113
| |
| | | 403.955
| |
| 276.106
| |
| | | 286.135
| |
| 195.575
| |
| | | 117.82
| |
| 80.531
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | | 7/33
| |
| | |
| |
| | |
| |
| | |
| |
| | | 403.445
| |
| 275.758
| |
| | | 288.175
| |
| 196.97
| |
| | | 115.27
| |
| 78.788
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 25/118
| |
| | | 402.957
| |
| 275.424
| |
| | | 290.129
| |
| 198.305
| |
| | | 112.828
| |
| 77.119
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 18/85
| |
| | |
| |
| | | 402.767
| |
| 275.294
| |
| | | 290.887
| |
| 198.8235
| |
| | | 111.88
| |
| 76.471
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 29/137
| |
| | | 402.604
| |
| 275.1825
| |
| | | 291.5405
| |
| 199.27
| |
| | | 111.063
| |
| 75.912
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 11/52
| |
| | |
| |
| | |
| |
| | | 402.337
| |
| 275
| |
| | | 292.6085
| |
| 200
| |
| | | 109.728
| |
| 75
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 26/123
| |
| | | 402.039
| |
| 274.797
| |
| | | 293.79
| |
| 200.813
| |
| | | 108.241
| |
| 73.984
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 402.015
| |
| 274.78
| |
| | | 293.896
| |
| 200.88
| |
| | | 108.118
| |
| 73.9
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 15/71
| |
| | |
| |
| | | 401.8215
| |
| 274.648
| |
| | | 294.669
| |
| 201.4085
| |
| | | 107.152
| |
| 73.239
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 19/90
| |
| | | 401.524
| |
| 274.444
| |
| | | 295.829
| |
| 202.222
| |
| | | 105.664
| |
| 72.222
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | | 4\19
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 400.412
| |
| 273.684
| |
| | | 300.309
| |
| 205.263
| |
| | | 100.103
| |
| 68.421
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 399.692
| |
| 273.193
| |
| | | 303.1855
| |
| 207.2295
| |
| | | 96.507
| |
| 65.963
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 17/81
| |
| | | 399.176
| |
| 272.8395
| |
| | | 305.252
| |
| 208.642
| |
| | | 93.924
| |
| 64.1975
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 13/62
| |
| | |
| |
| | | 398.797
| |
| 272.551
| |
| | | 306.767
| |
| 209.677
| |
| | | 92.03
| |
| 62.903
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 22/105
| |
| | | 398.505
| |
| 272.381
| |
| | | 307.936
| |
| 210.476
| |
| | | 90.569
| |
| 61.905
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 9/43
| |
| | |
| |
| | |
| |
| | | 398.084
| |
| 272.093
| |
| | | 309.621
| |
| 211.628
| |
| | | 88.463
| |
| 60.465
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 23/110
| |
| | | 397.6815
| |
| 271.818
| |
| | | 311.229
| |
| 212.727
| |
| | | 86.4525
| |
| 59.091
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 14/67
| |
| | |
| |
| | | 397.423
| |
| 271.641
| |
| | | 312.261
| |
| 213.433
| |
| | | 85.162
| |
| 58.209
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 19/91
| |
| | | 397.1115
| |
| 271.429
| |
| | | 313.509
| |
| 214.286
| |
| | | 83.602
| |
| 57.143
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | | 5/24
| |
| | |
| |
| | |
| |
| | |
| |
| | | 396.241
| |
| 270.833
| |
| | | 316.9925
| |
| 216.667
| |
| | | 79.248
| |
| 54.167
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 16/77
| |
| | | 395.211
| |
| 270.13
| |
| | | 321.109
| |
| 219.4805
| |
| | | 74.102
| |
| 50.649
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 11/53
| |
| | |
| |
| | | 394.745
| |
| 269.811
| |
| | | 322.9735
| |
| 220.755
| |
| | | 71.772
| |
| 49.057
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 17/82
| |
| | | 394.308
| |
| 269.8512
| |
| | | 324.724
| |
| 221.951
| |
| | | 69.584
| |
| 47.561
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 6/29
| |
| | |
| |
| | |
| |
| | | 393.508
| |
| 268.9655
| |
| | | 327.923
| |
| 224.138
| |
| | | 65.585
| |
| 44.8275
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 13/63
| |
| | | 392.467
| |
| 265.254
| |
| | | 332.087
| |
| 226.984
| |
| | | 60.3795
| |
| 41.27
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 7/34
| |
| | |
| |
| | | 391.579
| |
| 267.647
| |
| | | 335.639
| |
| 229.412
| |
| | | 55.94
| |
| 38.235
| |
| | |
| |
| |-
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | | 8/39
| |
| | | 390.145
| |
| 266.667
| |
| | | 341.3765
| |
| 233.333
| |
| | | 48.768
| |
| 33.333
| |
| | |
| |
| |-
| |
| | | 1/5
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | |
| |
| | colspan="2" | 380.391
| |
| 260
| |
| | | 0
| |
| | |
| |
| |}
| |
| | |
| =Triple Bohlen-Pierce=
| |
| Proposed by [[Paul Erlich]], is the Triple Bohlen-Pierce Scale, or 39th root of 3. It approximates additional odd harmonics and can be used in a variety of ways, for both just intonation chords and harmonies, as standard Bohlen-Pierce scale interlocking three times with calm sounding quarter-tones, and for various JI modulations.
| |
| | |
| | |
| =Theory=
| |
| [http://www.huygens-fokker.org/bpsite/ Bohlen Pierce website]
| |
| | |
| [http://en.wikipedia.org/wiki/Bohlen-Pierce_scale Wikipedia Bohlen-Pierce scale]
| |
| | |
| [http://ziaspace.com/_academic/BP/ Bohlen-Pierce Scale Research] by Elaine Walker
| |
| | |
| Sword, Ronald. "Bohlen Pierce Scales for Guitar" IAAA Press, UK-USA. First Ed: May 2009.
| |
| | |
| [[Intervals of BP]]
| |
| | |
| =Physical instruments tuned to the BP scale=
| |
| [http://www.swordguitars.com/Sword_Bohlen_Pierce_acoustic.JPG Bohlen Pierce guitar]
| |
| | |
| [[microtonal clarinet#BPClarinets|Clarinets]]
| |
| | |
| Metallophone
| |
| | |
| Electronic Organ
| |
| | |
| Stredici
| |
| | |
| [http://bp.b0b.com/2016/01/tuning-a-kalimba-to-bp/ Kalimba] (Mbira)
| |
| | |
| [http://bp.b0b.com/2012/11/sierra-pedal-steel/ Pedal Steel Guitar]
| |
| | |
| =Compositions=
| |
| [https://soundcloud.com/overtoneshock/a-mean-little-voice-bohlen-pierce-or-13ed3 A Mean Little Voice] by Stephen Weigel
| |
| | |
| [http://chrisvaisvil.com/ask-for-it-over-the-ridge/ Ask For It] by [[Chris Vaisvil]]
| |
| | |
| Links to available music written in BP at [http://www.huygens-fokker.org/bpsite/references2.html#anchor81501 above website].
| |
| | |
| | |
| [http://www.seraph.it/XenoTunes4_files/Bohl-en%20Roll.mp3 Bohl-en Roll] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/004e4fe53f09c06fd5e709a08d53d0d5-4.html blog entry])
| |
| | |
| [http://www.jeanpierrepoulin.com/mp3/BPguitares.mp3 Bohlen-Pierce electric guitar improvisation] by [[Jean-Pierre Poulin]]
| |
| | |
| [http://www.ronsword.com/sounds/BP_stretched_chroma.mp3 Bohlen-Pierce "Stretched Chroma" Acoustic Improvisation] by [[Ron Sword]]
| |
| | |
| [http://www.kunstmusik.com/mp3/reminiscences.mp3 Reminiscences] by [http://www.kunstmusik.com/2010/03/08/reminiscences/ Steven Yi]
| |
| | |
| [http://www.jeanpierrepoulin.com/mp3/Roll%27n%27Peace.mp3 Roll'n'Peace] by [[Jean-Pierre Poulin]]
| |
| | |
| [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/01%20-%201.%20Bohlen-Pierce.mp3 Comets Over Flatland 1] by [[Randy Winchester]]
| |
| | |
| [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/02%20-%202.%20Bohlen-Pierce.mp3 Comets Over Flatland 2] by [[Randy Winchester]]
| |
| | |
| [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/03%20-%203.%20Bohlen-Pierce.mp3 Comets Over Flatland 3] by [[Randy Winchester]]
| |
| | |
| [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/04%20-%204.%20Bohlen-Pierce.mp3 Comets Over Flatland 4] by [[Randy Winchester]]
| |
|
| |
|
| [http://micro.soonlabel.com/bohlen-pierce/daily20110915_BP_strings-Bohlen-Pierce-Island.mp3 Bohlen-Pierce Island audio] by [[Chris Vaisvil]] | | == External links == |
| | | * [http://www.huygens-fokker.org/bpsite/ The Bohlen-Pierce Site] |
| [http://micro.soonlabel.com/bohlen-pierce/mesonic_atom_13th_root_of_3_BP_apx_X1.mp3 Mesonic Atom] by [[Chris Vaisvil]]
| | * [http://ziaspace.com/_academic/BP/ Bohlen-Pierce Scale Research] by [[Elaine Walker]] |
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| [http://micro.soonlabel.com/MOTU/Vaisvil-MOTU-8a-bending-the-rules-in-bohlen-pierce.mp3 Bending the Rules] by [[Chris Vaisvil]]
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| [http://micro.soonlabel.com/gene_ward_smith/Others/Hansen/Bohlen-Pierce%20Canon.mp3 Bohlen-Pierce Canon] by [http://www.youtube.com/watch?v=sd1b9Lh8iFA Kjell Hansen]. | |
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| [http://chrisvaisvil.com/bohlens-pierced-waltz/ Bohlen's Pierced Waltz] by [[Chris Vaisvil]]
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| [http://chrisvaisvil.com/the-complex-plane-bp-guitar/ The Complex Plane] by [[Chris Vaisvil]]
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| ==See also==
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| * [[Catalog of 3.5.7 subgroup rank two temperaments]]
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| [[Category:Bohlen-pierce]] | | [[Category:Bohlen–Pierce| ]] <!-- main article --> |
| [[Category:Bp]] | | [[Category:13-tone scales]] |
| [[Category:Edonoi]]
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| [[Category:Edt]]
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| [[Category:Equal]]
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| [[Category:Guitar]] | | [[Category:Guitar]] |
| [[Category:Listen]]
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| [[Category:Macrotonal]] | | [[Category:Macrotonal]] |
| [[Category:Scale]]
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| [[Category:Theory]]
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