Editing Bohlen–Pierce scale (section)

====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
|}