Editing Bohlen–Pierce scale (section)

==Other unusual tunings or scales==
Other non-octave tunings investigated by Bohlen<ref>Bohlen (1978). footnote 26, page 84.</ref> include twelve steps in the tritave, named [[A12 scale|A12]] by Enrique Moreno<ref>{{cite web |url=http://www.huygens-fokker.org/bpsite/otherscales.html |title=Other Unusual Scales |work=The Bohlen–Pierce Site |access-date=27 November 2012}} Cites: {{cite journal |last1=Moreno |first1=Enrique Ignacio |date=Dec 1995 |title=Embedding Equal Pitch Spaces and The Question of Expanded Chromas: An Experimental Approach |journal=Dissertation |pages=12–22 |publisher=Stanford University}}</ref> and based on the 4:7:10 chord {{audio|A12 4 7 10 on C.mid|Play}}, seven steps in the octave ([[7-tet]]) or similar 11 steps in the tritave, and eight steps in the octave, based on 5:7:9 {{audio|5 7 9 chord on E.mid|Play}} and of which only the just version would be used. Additionally, the pentave can be divided into eight steps which approximates chords of the form 5:9:13:17:21:25.<ref>"[http://www.huygens-fokker.org/bpsite/otherscales.html Other Unusual Scales]", ''The Bohlen–Pierce Site''. Retrieved 27 November 2012. Cites: Bohlen (1978). pp. 76–86.</ref> The [[833 cents scale|Bohlen 833 cents scale]] is based on the [[Fibonacci sequence]], although it was created from [[combination tone]]s, and contains a complex network of harmonic relations due to the inclusion of coinciding harmonics of stacked 833 cent intervals. For example, "step 10 turns out to be identical with the octave (1200 cents) to the base tone, at the same time featuring the [[Golden Ratio]] to step 3".<ref>{{cite web |url=http://www.huygens-fokker.org/bpsite/833cent.html |title=An 833 Cents Scale |last1=Bohlen |first1=Heinz |work=The Bohlen–Pierce Site |access-date=27 November 2012}}</ref>

Alternate scales may be specified by indicating the size of equal tempered steps, for example [[Wendy Carlos]]' 78-cent [[alpha scale]] and 63.8-cent [[beta scale]], and Gary Morrison's 88-cent scale (13.64 steps per octave or 14 per 1232-cent stretched octave).<ref>{{cite book |last=Sethares |first=William |author-link=William Sethares |title=Tuning, Timbre, Spectrum, Scale |year=2004 |isbn=1-85233-797-4 |page=60}}</ref> This gives the alpha scale 15.39 steps per octave and the beta scale 18.75 steps per octave.<ref>{{cite AV media notes |title=Beauty in the Beast |title-link=Beauty in the Beast |others=Wendy Carlos |year=2000 |orig-year=1986 |chapter=Liner notes |first=Wendy |last=Carlos |author-link=Wendy Carlos |type=CD |publisher=ESD |id=81552}}</ref>

===Expansions===
====39-tone equal division of the tritave====
Paul Erlich proposed dividing each step of the Bohlen–Pierce into thirds so that the tritave is divided into 39 equal steps instead of 13 equal steps. The scale, which can be viewed as three evenly staggered Bohlen-Pierce scales, gives additional odd harmonics. The 13-step scale hits the odd harmonics 3:1; 5:3, 7:3; 7:5, 9:5; 9:7, and 15:7; while the 39-step scale includes all of those and many more (11:5, 13:5; 11:7, 13:7; 11:9, 13:9; 13:11, 15:11, 21:11, 25:11, 27:11; 15:13, 21:13, 25:13, 27:13, 33:13, and 35:13), while still missing almost all of the even harmonics (including 2:1; 3:2, 5:2; 4:3, 8:3; 6:5, 8:5; 9:8, 11:8, 13:8, and 15:8). The size of this scale is about 25 equal steps to a ratio slightly larger than an octave, so each of the 39 equal steps is slightly smaller than half of one of the 12 equal steps of the standard scale.<ref>{{cite web |url=http://www.huygens-fokker.org/bpsite/scales.html |title=BP Scale Structures |work=The Bohlen–Pierce Site |access-date=27 November 2012}}</ref>
{| class="wikitable sortable mw-collapsible mw-collapsed"
|-
! Number of equally-tempered steps !! Equally-tempered interval !! Size of equally-tempered interval (cents) !! Justly-intoned interval !! Size of justly-intoned interval (cents) !! Error (cents)
|-
| 91 || 12.9802 || 4437.90 || 13/1 || 4440.53 || -2.63
|-
| 85 || 10.9617 || 4145.29 || 11/1 || 4151.32 || -6.03
|-
| 69 || 6.9845 || 3365.00 || 7/1 || 3368.83 || -3.83
|-
| 57 || 4.9812 || 2779.78 || 5/1 || 2786.31 || -6.53
|-
| 49 || 3.9761 || 2389.64 || 4/1 || 2400.00 || -10.36
|-
| 39 || 3.0000 || 1901.96 || 3/1 || 1901.96 || 0.00
|-
| rowspan="4" |38
| rowspan="4" |2.9167
| rowspan="4" |1853.19
| 225/77 || 1856.39 || -3.21
|-
| 35/12 || 1853.18 || 0.00
|-
| 32/11 || 1848.68 || 4.50
|-
| 189/65 || 1847.85 || 5.34
|-
| 37 || 2.8357 || 1804.42 || 99/35 || 1800.09 || 4.33
|-
| rowspan="3" | 36 || rowspan="3" | 2.7569 || rowspan="3" | 1755.65 || 36/13 || 1763.38 || -7.73
|-
| 135/49 || 1754.53 || 1.12
|-
| 11/7 || 1751.32 || 4.33
|-
| 35 || 2.6803 || 1706.88 || 35/13 || 1714.61 || -7.73
|-
| 34 || 2.6059 || 1658.11 || 13/5 || 1654.21 || 3.90
|-
| rowspan="2" |33
| rowspan="2" |2.5335
| rowspan="2" |1609.35
| 63/25 || 1600.11 || 9.24
|-
| 33/13 || 1612.75 || -3.40
|-
| 32 || 2.4631 || 1560.58 || 27/11 || 1554.55 || 6.03
|-
| rowspan="2" |31
| rowspan="2" |2.3947
| rowspan="2" |1511.81
| 12/5 || 1515.64 || -3.83
|-
| 117/49 || 1506.79 || 5.02
|-
| 30 || 2.3282 || 1463.04 || 7/3 || 1466.87 || -3.83
|-
| rowspan="2" | 29 || rowspan="2" | 2.2635 || rowspan="2" | 1414.27 || 25/11 || 1421.31 || -7.04
|-
| 147/65 || 1412.77 || 1.51
|-
| 28 || 2.2006 || 1365.51 || 11/5 || 1365.00 || 0.50
|-
| 27 || 2.1395 || 1316.74 || 15/7 || 1319.44 || -2.70
|-
| 26 || 2.0801 || 1267.97 || 27/13 || 1265.34 || 2.63
|-
| 25 || 2.0223 || 1219.20 || 99/49 || 1217.58 || 1.63
|-
| 24 || 1.9661 || 1170.43 || 49/25 || 1165.02 || 5.41
|-
| 23 || 1.9115 || 1121.67 || 21/11 || 1119.46 || 2.20
|-
| 22 || 1.8584 || 1072.90 || 13/7 || 1071.70 || 1.20
|-
| 21 || 1.8068 || 1024.13 || 9/5 || 1017.60 || 6.53
|-
| rowspan="2" | 20 || rowspan="2" | 1.7566 || rowspan="2" | 975.36 || 135/77 || 972.03 || 3.33
|-
| 7/4 || 968.83 || 6.54
|-
| rowspan="2" | 19 || rowspan="2" | 1.7078 || rowspan="2" | 926.59 || 12/7 || 933.13 || -6.54
|-
| 77/45 || 929.92 || -3.33
|-
| 18 || 1.6604 || 877.83 || 5/3 || 884.36 || -6.53
|-
| 17 || 1.6143 || 829.06 || 21/13 || 830.25 || -1.20
|-
| 16 || 1.5694 || 780.29 || 11/7 || 782.49 || -2.20
|-
|15
|1.5258
|731.52
| 75/49 || 736.93 || -5.41
|-
| 14 || 1.4835 || 682.75 || 49/33 || 684.38 || -1.63
|-
| 13 || 1.4422 || 633.99 || 13/9 || 636.62 || -2.63
|-
|12
|1.4022
|585.22
| 7/5 || 582.51 || 2.70
|-
| 11 || 1.3632 || 536.45 || 15/11 || 536.95 || -0.50
|-
| rowspan="2" | 10 || rowspan="2" | 1.3254 || rowspan="2" | 487.68 || 65/49 || 489.19 || -1.51
|-
| 33/25 || 480.65 || 7.04
|-
| 9 || 1.2886 || 438.91 || 9/7 || 435.08 || 3.83
|-
| rowspan="2" | 8 || rowspan="2" | 1.2528 || rowspan="2" | 390.14 || 49/39 || 395.17 || -5.02
|-
| 5/4 || 386.31 || 3.83
|-
|7
|1.2180
|341.38
| 11/9 || 347.41 || -6.03
|-
| rowspan="2" | 6 || rowspan="2" | 1.1841 || rowspan="2" | 292.61 || 13/11 || 289.21 || 3.40
|-
| 25/21 || 301.85 || -9.24
|-
|5
|1.1512
|243.84
| 15/13 || 247.74 || -3.90
|-
| 4 || 1.1193 || 195.07 || 39/35 || 187.34 || 7.73
|-
| rowspan="3" |3
| rowspan="3" |1.0882
| rowspan="3" |146.30
|12/11
|150.64
| -4.33
|-
|49/45
|147.43
| -1.12
|-
|13/12
|138.57
|7.73
|-
|2
|1.0580
|97.54
|35/33
|101.87
| -4.33
|-
| rowspan="4" |1
| rowspan="4" |1.0286
| rowspan="4" |48.77
|65/63
|54.11
| -5.34
|-
|33/32
|53.27
| -4.50
|-
|36/35
|48.77
|0.00
|-
|77/75
|45.56
|3.21
|-
|0
|1.0000
|0.00
|1/1
|0.00
|0.00
|}

====65-tone equal division of the tritave====
Dividing each step of the Bohlen–Pierce scale into fifths (so that the tritave is divided into 65 steps) results in a very accurate octave (41 steps) and perfect fifth (24 steps), as well as approximations for other just intervals. The scale is practically identical to [[41 equal temperament|41-tone equal division of the octave]] except that each step is slightly smaller (less than a hundredth of a cent per step).
{| class="wikitable sortable mw-collapsible mw-collapsed"
|-
! Number of equally-tempered steps !! Equally-tempered interval !! Size of equally-tempered interval (cents) !! Justly-intoned interval !! Size of justly-intoned interval (cents) !! Error (cents)
|-
| 65 || 3.0000 || 1901.96 || 3/1 || 1901.9550 || 0.00
|-
| 64 || 2.9497 || 1872.69 || 144/49 || 1866.2582 || 6.44
|-
| 63 || 2.9003 || 1843.43 || 32/11 || 1848.6821 || -5.25
|-
| 62 || 2.8517 || 1814.17 || 20/7 || 1817.4878 || -3.32
|-
| 61 || 2.8039 || 1784.91 || 14/5 || 1782.5122 || 2.40
|-
| rowspan="2" | 60 || rowspan="2" | 2.7569 || rowspan="2" | 1755.65 || 135/49 || 1754.5269 || 1.12
|-
| 11/4 || 1751.3179 || 4.33
|-
| 59 || 2.7107 || 1726.39 || 27/10 || 1719.5513 || 6.84
|-
| 58 || 2.6653 || 1697.13 || 8/3 || 1698.0450 || -0.92
|-
| 57 || 2.6206 || 1667.87 || 21/8 || 1670.7809 || -2.91
|-
| 56 || 2.5767 || 1638.61 || 18/7 || 1635.0841 || 3.52
|-
| 55 || 2.5335 || 1609.35 || 81/32 || 1607.8200 || 1.53
|-
| 54 || 2.4910 || 1580.09 || 5/2 || 1586.3137 || -6.23
|-
| 53 || 2.4493 || 1550.82 || 27/11 || 1554.5471 || -3.72
|-
| 52 || 2.4082 || 1521.56 || 12/5 || 1515.6413 || 5.92
|-
| 51 || 2.3679 || 1492.30 || 64/27 || 1494.1350 || -1.83
|-
| 50 || 2.3282 || 1463.04 || 7/3 || 1466.8709 || -3.83
|-
| 49 || 2.2892 || 1433.78 || 16/7 || 1431.1741 || 2.61
|-
| 48 || 2.2508 || 1404.52 || 9/4 || 1403.9100 || 0.61
|-
| 47 || 2.2131 || 1375.26 || 20/9 || 1382.4037 || -7.14
|-
| 46 || 2.1760 || 1346.00 || 24/11 || 1350.6371 || -4.64
|-
| 45 || 2.1395 || 1316.74 || 15/7 || 1319.4428 || -2.70
|-
| 44 || 2.1037 || 1287.48 || 21/10 || 1284.4672 || 3.01
|-
| 43 || 2.0684 || 1258.22 || 33/16 || 1253.2729 || 4.94
|-
| 42 || 2.0337 || 1228.96 || 55/27 || 1231.7667 || -2.81
|-
| 41 || 1.9996 || 1199.69 || 2/1 || 1200.0000 || -0.31
|-
| 40 || 1.9661 || 1170.43 || 49/25 || 1165.0244 || 5.41
|-
| 39 || 1.9332 || 1141.17 || 27/14 || 1137.0391 || 4.13
|-
| 38 || 1.9008 || 1111.91 || 40/21 || 1115.5328 || -3.62
|-
| 37 || 1.8689 || 1082.65 || 15/8 || 1088.2687 || -5.62
|-
| 36 || 1.8376 || 1053.39 || 11/6 || 1049.3629 || 4.03
|-
| 35 || 1.8068 || 1024.13 || 9/5 || 1017.5963 || 6.53
|-
| 34 || 1.7765 || 994.87 || 16/9 || 996.0900 || -1.22
|-
| 33 || 1.7468 || 965.61 || 7/4 || 968.8259 || -3.22
|-
| 32 || 1.7175 || 936.35 || 12/7 || 933.1291 || 3.22
|-
| 31 || 1.6887 || 907.09 || 27/16 || 905.8650 || 1.22
|-
| 30 || 1.6604 || 877.83 || 5/3 || 884.3587 || -6.53
|-
| 29 || 1.6326 || 848.56 || 18/11 || 852.5921 || -4.03
|-
| 28 || 1.6052 || 819.30 || 8/5 || 813.6863 || 5.62
|-
| 27 || 1.5783 || 790.04 || 63/40 || 786.4222 || 3.62
|-
| 26 || 1.5518 || 760.78 || 14/9 || 764.9159 || -4.13
|-
| 25 || 1.5258 || 731.52 || 32/21 || 729.2191 || 2.30
|-
| 24 || 1.5003 || 702.26 || 3/2 || 701.9550 || 0.31
|-
| rowspan="2" | 23 || rowspan="2" | 1.4751 || rowspan="2" | 673.00 || 81/55 || 670.1883 || 2.81
|-
| 72/49 || 666.2582 || 6.74
|-
| 22 || 1.4504 || 643.74 || 16/11 || 648.6821 || -4.94
|-
| 21 || 1.4261 || 614.48 || 10/7 || 617.4878 || -3.01
|-
| 20 || 1.4022 || 585.22 || 7/5 || 582.5122 || 2.70
|-
| 19 || 1.3787 || 555.96 || 11/8 || 551.3179 || 4.64
|-
| 18 || 1.3556 || 526.70 || 27/20 || 519.5513 || 7.14
|-
| 17 || 1.3329 || 497.43 || 4/3 || 498.0450 || -0.61
|-
| 16 || 1.3105 || 468.17 || 21/16 || 470.7809 || -2.61
|-
| 15 || 1.2886 || 438.91 || 9/7 || 435.0841 || 3.83
|-
| rowspan="2" | 14 || rowspan="2" | 1.2670 || rowspan="2" | 409.65 || 80/63 || 413.5778 || -3.93
|-
| 81/64 || 407.8200 || 1.83
|-
| 13 || 1.2457 || 380.39 || 5/4 || 386.3137 || -5.92
|-
| 12 || 1.2249 || 351.13 || 11/9 || 347.4079 || 3.72
|-
| 11 || 1.2043 || 321.87 || 6/5 || 315.6413 || 6.23
|-
| 10 || 1.1841 || 292.61 || 32/27 || 294.1350 || -1.53
|-
| 9 || 1.1643 || 263.35 || 7/6 || 266.8709 || -3.52
|-
| 8 || 1.1448 || 234.09 || 8/7 || 231.1741 || 2.91
|-
| 7 || 1.1256 || 204.83 || 9/8 || 203.9100 || 0.92
|-
| 6 || 1.1067 || 175.57 || 10/9 || 182.4037 || -6.84
|-
| rowspan="2" | 5 || rowspan="2" | 1.0882 || rowspan="2" | 146.30 || 12/11 || 150.6371 || -4.33
|-
| 49/45 || 147.4281 || -1.12
|-
| rowspan="2" | 4 || rowspan="2" | 1.0699 || rowspan="2" | 117.04 || 15/14 || 119.4428 || -2.40
|-
| 16/15 || 111.7313 || 5.31
|-
| 3 || 1.0520 || 87.78 || 21/20 || 84.4672 || 3.32
|-
| rowspan="2" | 2 || rowspan="2" | 1.0344 || rowspan="2" | 58.52 || 28/27 || 62.9609 || -4.44
|-
| 33/32 || 53.2729 || 5.25
|-
| rowspan="5" | 1 || rowspan="5" | 1.0170 || rowspan="5" | 29.26 || 49/48 || 35.6968 || -6.44
|-
| 50/49 || 34.9756 || -5.71
|-
| 55/54 || 31.7667 || -2.51
|-
| 56/55 || 31.1943 || -1.93
|-
| 64/63 || 27.2641 || 2.00
|-
| 0 || 1.0000 || 0.00 || 1/1 || 0.0000 || 0.00
|}