Intervals of BP: Difference between revisions

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This page gives the intervals of the equal tempered [[BP|Bohlen-Pierce]] scale -- the 13th root of 3, along with some nearby just ratios for comparison.
This page lists the first 42 intervals of the [[Bohlen–Pierce]] scale in both its [[just intonation]] (JI) version and its [[Equal-step tuning|equal-tempered]] (ET) version, [[13edt]].
 
In the "nearby JI intervals, octave-reduced" columns, the intervals from the original just BP scale are in bold. Errors are sorted by ascending absolute value.


{| class="wikitable"
{| class="wikitable"
|-
|-
! | degree
! rowspan="2" | Degree
! | cents value
! colspan="2" | Just intonation
! | nearby frequency ratios
! colspan="2" | Equal-tempered
! | cents value
! colspan="3" | Nearby JI intervals, [[Octave reduction|octave-reduced]]
! | difference between BP interval & just interval
|-
|-
! colspan="3" | first octave and first tritave...
! Ratio
| |
! [[Cent]]s
| |
! [[Hekt]]s
! Cents
! Ratio
! Cents
! Error in 13edt (¢)
|-
|-
| | 0
| 0
| | 0
| 1/1
| | 1/1
| 0
| | 0
| 0
| | 0
| 0
| 1/1
| 0
| 0
|-
|-
| | 1
| 1
| | 146.3
| 27/25
| | 12/11
| 133.238
 
| 100
13/12
| 146.304
 
| 12/11<br />13/12<br />'''27/25'''
27/25
| 150.6<br />138.6<br />'''133.2'''
| | 150.6
| −4.3<br />+7.7<br />'''+13.1'''
 
138.6
 
133.2
| | -4.3
 
7.7
 
13.1
|-
|-
| | 2
| 2
| | 292.6
| 25/21
| | 32/27
| 301.847
 
| 200
13/11
| 292.608
 
| 32/27<br />13/11<br />19/16<br />'''25/21'''
19/16
| 294.1<br />289.2<br />297.5<br />'''301.8'''
 
| −1.5<br />+3.4<br />−4.9<br />'''−9.2'''
25/21
| | 294.1
 
289.2
 
297.5
 
301.8
| | -1.5
 
3.4
 
-4.9
 
-9.2
|-
|-
| | 3
| 3
| | 438.9
| 9/7
| | 9/7
| 435.084
| | 435.1
| 300
| | 3.8
| 438.913
| '''9/7'''
| '''435.1'''
| '''+3.8'''
|-
|-
| | 4
| 4
| | 585.2
| 7/5
| | 7/5
| 582.512
 
| 400
45/32
| 585.217
| | 582.5
| '''7/5'''<br />45/32
 
| '''582.5'''<br />590.2
590.2
| '''+2.7'''<br />−5.0
| | 2.7
 
-5.0
|-
|-
| | 5
| 5
| | 731.5
| 75/49
| | 32/21
| 736.931
 
| 500
26/17
| 731.521
 
| 32/21<br />26/17<br />'''75/49'''<br />49/32<br />20/13
75/49
| 729.2<br />735.6<br />'''736.9'''<br />737.7<br />745.8
 
| +2.3<br />−4.1<br />'''−5.4'''<br />−6.2<br />−14.3
49/32
 
20/13
| | 729.2
 
735.6
 
736.9
 
737.7
 
745.8
| | 2.3
 
-4.1
 
-5.4
 
-6.2
 
-14.3
|-
|-
| | 6
| 6
| | 877.8
| 5/3
| | 5/3
| 884.359
| | 884.4
| 600
| | -6.6
| 877.825
| '''5/3'''
| '''884.4'''
| '''−6.5'''
|-
|-
| | 7
| 7
| | 1024.1
| 9/5
| | 38/21
| 1017.596
 
| 700
29/16
| 1024.130
 
| 38/21<br />29/16<br />'''9/5'''<br />20/11
9/5
| 1026.7<br />1029.6<br />'''1017.6'''<br />1035.0
 
| −2.6<br />−5.5<br />'''+6.5'''<br />−10.9
20/11
| | 1026.7
 
1029.6
 
1017.6
 
1035.0
| | -2.6
 
-5.5
 
6.5
 
-10.9
|-
|-
| | 8
| 8
| | 1170.4
| 49/25
| | 49/25
| 1165.024
| | 1165.0
| 800
| | 5.4
| 1170.434
| '''49/25'''
| '''1165.0'''
| '''+5.4'''
|-
|-
! colspan="3" | second octave...
| 9
| |  
| 15/7
| |
| 1319.443
| 900
| 1316.738
| '''15/14'''<br />16/15<br />14/13
| '''119.4'''<br />111.7<br />128.3
| '''−2.7'''<br />+5.0<br />−11.6
|-
|-
| | 9
| 10
| | 116.7
| 7/3
| | 15/14
| 1466.871
 
| 1000
16/15
| 1463.042
 
| '''7/6'''<br />22/19
14/13
| '''266.9'''<br />253.8
| | 119.4
| '''−3.8'''<br />+9.2
 
111.7
 
128.3
| | -2.7
 
5.0
 
-11.6
|-
|-
| | 10
| 11
| | 263.0
| 63/25
| | 7/6
| 1600.108
 
| 1100
22/19
| 1609.347
| | 266.9
| 19/15<br />81/64<br />24/19<br />14/11<br />'''63/50'''
 
| 409.2<br />407.8<br />404.4<br />417.5<br />'''400.1'''
253.8
| +0.1<br />+1.5<br />+4.9<br />−8.2<br />'''+9.2'''
| | -3.9
 
9.2
|-
|-
| | 11
| 12
| | 409.3
| 25/9
| | 19/15
| 1768.717
 
| 1200
81/64
| 1755.651
 
| 11/8<br />18/13<br />26/19<br />'''25/18'''
24/19
| 551.3<br />563.4<br />543.0<br />'''568.7'''
 
| +4.4<br />−7.7<br />+12.7<br />'''−13.1'''
14/11
 
63/50
| | 409.2
 
407.8
 
404.4
 
417.5
 
400.1
| | .1
 
1.5
 
4.9
 
-8.2
 
9.2
|-
|-
| | 12
| 13
| | 555.7
| 3/1
| | 11/8
| 1901.955
 
| 1300
18/13
| 1901.955
 
| '''3/2'''
26/19
| '''702.0'''
 
| '''0'''
25/18
| | 551.3
 
563.4
 
543.0
 
568.7
| | 4.4
 
-7.7
 
12.7
 
-13.0
|-
|-
! colspan="3" | second tritave...
| 14
| |  
| 81/25
| |
| 2035.193
| 1400
| 2048.259
| 18/11<br />13/8<br />'''81/50'''
| 852.6<br />840.5<br />'''835.2'''
| −4.3<br />+7.8<br />'''+13.1'''
|-
|-
| | 13
| 15
| | 702.0
| 25/7
| | 3/2
| 2203.802
| | 702.0
| 1500
| | 0
| 2194.563
| 16/9<br />'''25/14'''
| 996.1<br />'''1003.8'''
| −1.5<br />'''−9.2'''
|-
|-
| | 14
| 16
| | 848.3
| 27/7
| | 18/11
| 2337.039
 
| 1600
13/8
| 2340.868
| | 852.6
| '''27/14'''
 
| '''1137.0'''
840.5
| '''+3.8'''
| | -4.3
 
7.8
|-
|-
| | 15
| 17
| | 994.6
| 21/5
| | 16/9
| 2484.467
| | 996.1
| 1700
| | -1.5
| 2487.172
| 20/19<br />'''21/20'''<br />19/18<br />22/21
| 88.8<br />'''84.5'''<br />93.6<br />80.5
| −1.6<br />'''+2.7'''<br />−6.4<br />+6.7
|-
|-
| | 16
| 18
| | 1140.9
| 225/49
| | 27/14
| 2638.886
| | 1137.0
| 1800
| | 3.9
| 2633.476
| 8/7<br />'''225/196'''<br />15/13
| 231.2<br />'''238.9'''<br />247.7
| +2.3<br />'''−5.4'''<br />−14.2
|-
|-
! colspan="3" | third octave...
| 19
| |  
| 5/1
| |
| 2786.314
| 1900
| 2779.780
| '''5/4'''<br />26/21
| '''386.3'''<br />369.7
| '''−6.5'''<br />+10.1
|-
|-
| | 17
| 20
| | 87.2
| 27/5
| | 20/19
| 2919.551
 
| 2000
21/20
| 2926.085
 
| 19/14<br />'''27/20'''<br />15/11
19/18
| 528.7<br />'''519.6'''<br />537.0
 
| −2.6<br />'''+6.5'''<br />−10.9
22/21
| | 88.8
 
84.5
 
93.6
 
80.5
| | -1.6
 
2.7
 
-6.4
 
6.7
|-
|-
| | 18
| 21
| | 233.5
| 147/25
| | 8/7
| 3066.979
 
| 2100
15/13
| 3072.389
| | 231.2
| 28/19<br />25/17<br />'''147/100'''<br />22/15
 
| 671.3<br />667.7<br />'''667.0'''<br />663.0
247.7
| +1.1<br />+4.7<br />'''+5.4'''<br />+9.4
| | 2.3
 
-14.2
|-
|-
| | 19
| 22
| | 379.8
| 45/7
| | 5/4
| 3221.398
 
| 2200
26/21
| 3218.693
| | 386.3
| '''45/28'''<br />8/5<br />21/13
 
| '''821.4'''<br />813.7<br />830.3
369.7
| '''−2.7'''<br />+5.0<br />−11.6
| | -6.5
 
10.1
|-
|-
| | 20
| 23
| | 526.1
| 7/1
| | 19/14
| 3368.826
 
| 2300
27/20
| 3364.997
 
| '''7/4'''<br />26/15
15/11
| '''968.8'''<br />952.3
| | 528.7
| '''−3.8'''<br />+12.7
 
519.6
 
537.0
| | -2.6
 
6.5
 
-10.9
|-
|-
| | 21
| 24
| | 672.4
| 189/25
| | 28/19
| 3502.063
 
| 2400
25/17
| 3511.302
 
| 19/10<br />40/21<br />36/19<br />21/11<br />'''189/100'''
22/15
| 1111.2<br />1115.5<br />1106.4<br />1119.5<br />'''1102.1'''
| | 671.3
| +0.1<br />−4.2<br />+4.9<br />−8.2<br />'''+9.2'''
 
667.7
 
663.0
| | 1.1
 
4.7
 
9.4
|-
|-
| | 22
| 25
| | 818.7
| 25/3
| | 8/5
| 3670.672
 
| 2500
21/13
| 3657.606
| | 813.7
| 33/32<br />'''25/24'''
 
| 53.3<br />'''70.7'''
830.3
| +4.3<br />'''−13.1'''
| | 5.0
 
-11.6
|-
|-
| | 23
| 26
| | 965.0
| 9/1
| | 7/4
| 3803.910
 
| 2600
26/15
| 3803.910
| | 968.8
| '''9/8'''
 
| '''203.9'''
952.3
| '''0'''
| | -3.8
 
12.7
|-
| | 24
| | 1111.3
| | 19/10
 
40/21
 
36/19
 
21/11
| | 1111.2
 
1115.5
 
1106.4
 
1119.5
| | 0.1
 
-4.2
 
4.9
 
-8.2
|-
|-
! colspan="3" | fourth octave...
| 27
| |  
| 243/25
| |
| 3937.148
| 2700
| 3950.214
| 11/9<br />27/22<br />'''243/200'''
| 347.4<br />354.5<br />337.1
| +2.8<br />−4.3<br />'''+13.1'''
|-
|-
| | 25
| 28
| | 57.6
| 75/7
| | 33/32
| 4105.757
| | 53.3
| 2800
| | 4.3
| 4096.518
| 4/3<br />'''75/56'''
| 498.0<br />'''505.8'''
| −1.5<br />'''−9.2'''
|-
|-
! colspan="3" | third tritave...
| 29
| |  
| 81/7
| |
| 4238.994
| 2900
| 4242.823
| '''81/56'''<br />36/25
| '''639.0'''<br />631.3
| '''+3.8'''<br />+11.5
|-
|-
| | 26
| 30
| | 203.9
| 63/5
| | 9/8
| 4386.422
| | 203.9
| 3000
| | 0
| 4389.127
| 30/19<br />'''63/40'''<br />128/81<br />19/12<br />11/7<br />100/63
| 790.8<br />'''786.4'''<br />792.2<br />795.6<br />782.5<br />799.9.
| −1.7<br />'''2.7'''<br />−3.1<br />−6.5<br />+6.6<br />−10.8
|-
|-
| | 27
| 31
| | 350.2
| 675/49
| |  
| 4540.841
| |  
| 3100
| |
| 4535.431
| 12/7<br />'''675/392'''<br />19/11
| 933.1<br />'''940.8'''<br />946.2
| +2.3<br />'''−5.4'''<br />+10.8
|-
|-
| | 28
| 32
| | 496.5
| 15/1
| |  
| 4688.269
| |  
| 3200
| |
| 4681.735
| 28/15<br />'''15/8'''<br />13/7
| 1080.6<br />'''1088.3'''<br />1071.7
| +1.1<br />'''−6.5'''<br />+10.0
|-
|-
| | 29
| 33
| | 642.8
| 81/5
| |  
| 4821.506
| |  
| 3300
| |
| 4828.040
| 64/63<br />'''81/80'''
| 27.3<br />'''21.5'''
| +0.7<br />'''+6.5'''
|-
|-
| | 30
| 34
| | 789.1
| 441/25
| |  
| 4968.934
| |  
| 3400
| |
| 4974.344
| 21/19<br />'''441/400'''<br />10/9<br />11/10
| 173.3<br />'''168.9'''<br />182.4<br />165.0
| +1.0<br />'''+5.4'''<br />−8.1<br />+9.3
|-
|-
| | 31
| 35
| | 935.4
| 135/7
| |  
| 5123.353
| |  
| 3500
| |
| 5120.648
| '''135/112'''<br />6/5
| '''323.4'''<br />315.6
| '''−2.7'''<br />+5.0
|-
|-
| | 32
| 36
| | 1081.7
| 21/1
| |  
| 5270.781
| |  
| 3600
| |
| 5266.952
| 17/13<br />'''21/16'''<br />98/75<br />64/49<br />13/10
| 464.4<br />'''470.8'''<br />463.1<br />462.3<br />454.2<br />
| +2.5<br />'''−3.8'''<br />+3.9<br />+4.6<br />+12.7
|-
|-
| colspan="3" style="text-align:center;" | '''fifth octave...'''
| 37
| |
| 567/25
| |
| 5404.018
| 3700
| 5413.257
| 64/45<br />10/7<br />567/400
| 609.8<br />617.5<br />'''604.0'''
| +3.5<br />−4.2<br />'''+9.2'''
|-
|-
| | 33
| 38
| | 28.0
| 25/1
| |  
| 5572.627
| |  
| 3800
| |
| 5559.561
| 14/9<br />'''25/16'''
| 764.9<br />'''772.6'''
| −5.3<br />'''−13.1'''
|-
|-
| |  
| 39
| |  
| 27/1
| |  
| 5705.865
| |
| 3900
| |
| 5705.865
| '''27/16'''
| '''905.9'''
| '''0'''
|-
|-
| |  
| 40
| |  
| 729/25
| |  
| 5839.103
| |
| 4000
| |
| 5852.169
| 11/6<br />'''729/400'''
| 1049.4<br />'''1039.1'''
| +2.8<br />'''+13.1'''
|-
|-
| |
| 41
| |
| 225/7
| |
| 6007.712
| |
| 4100
| |
| 5998.473
|-
| 1/1<br />'''225/224'''
| |
| 0.0<br />'''7.7'''
| |
| −1.5<br />'''+9.2'''
| |
| |
| |
|-
| |
| |
| |
| |
| |
|}
|}
[[Category:Bohlen–Pierce]]
[[Category:Lists of intervals]]
[[Category:Tables]]

Latest revision as of 15:44, 31 July 2025

This page lists the first 42 intervals of the Bohlen–Pierce scale in both its just intonation (JI) version and its equal-tempered (ET) version, 13edt.

In the "nearby JI intervals, octave-reduced" columns, the intervals from the original just BP scale are in bold. Errors are sorted by ascending absolute value.

Degree Just intonation Equal-tempered Nearby JI intervals, octave-reduced
Ratio Cents Hekts Cents Ratio Cents Error in 13edt (¢)
0 1/1 0 0 0 1/1 0 0
1 27/25 133.238 100 146.304 12/11
13/12
27/25 		150.6
138.6
133.2 		−4.3
+7.7
+13.1
2 25/21 301.847 200 292.608 32/27
13/11
19/16
25/21 		294.1
289.2
297.5
301.8 		−1.5
+3.4
−4.9
−9.2
3 9/7 435.084 300 438.913 9/7 435.1 +3.8
4 7/5 582.512 400 585.217 7/5
45/32 		582.5
590.2 		+2.7
−5.0
5 75/49 736.931 500 731.521 32/21
26/17
75/49
49/32
20/13 		729.2
735.6
736.9
737.7
745.8 		+2.3
−4.1
−5.4
−6.2
−14.3
6 5/3 884.359 600 877.825 5/3 884.4 −6.5
7 9/5 1017.596 700 1024.130 38/21
29/16
9/5
20/11 		1026.7
1029.6
1017.6
1035.0 		−2.6
−5.5
+6.5
−10.9
8 49/25 1165.024 800 1170.434 49/25 1165.0 +5.4
9 15/7 1319.443 900 1316.738 15/14
16/15
14/13 		119.4
111.7
128.3 		−2.7
+5.0
−11.6
10 7/3 1466.871 1000 1463.042 7/6
22/19 		266.9
253.8 		−3.8
+9.2
11 63/25 1600.108 1100 1609.347 19/15
81/64
24/19
14/11
63/50 		409.2
407.8
404.4
417.5
400.1 		+0.1
+1.5
+4.9
−8.2
+9.2
12 25/9 1768.717 1200 1755.651 11/8
18/13
26/19
25/18 		551.3
563.4
543.0
568.7 		+4.4
−7.7
+12.7
−13.1
13 3/1 1901.955 1300 1901.955 3/2 702.0 0
14 81/25 2035.193 1400 2048.259 18/11
13/8
81/50 		852.6
840.5
835.2 		−4.3
+7.8
+13.1
15 25/7 2203.802 1500 2194.563 16/9
25/14 		996.1
1003.8 		−1.5
−9.2
16 27/7 2337.039 1600 2340.868 27/14 1137.0 +3.8
17 21/5 2484.467 1700 2487.172 20/19
21/20
19/18
22/21 		88.8
84.5
93.6
80.5 		−1.6
+2.7
−6.4
+6.7
18 225/49 2638.886 1800 2633.476 8/7
225/196
15/13 		231.2
238.9
247.7 		+2.3
−5.4
−14.2
19 5/1 2786.314 1900 2779.780 5/4
26/21 		386.3
369.7 		−6.5
+10.1
20 27/5 2919.551 2000 2926.085 19/14
27/20
15/11 		528.7
519.6
537.0 		−2.6
+6.5
−10.9
21 147/25 3066.979 2100 3072.389 28/19
25/17
147/100
22/15 		671.3
667.7
667.0
663.0 		+1.1
+4.7
+5.4
+9.4
22 45/7 3221.398 2200 3218.693 45/28
8/5
21/13 		821.4
813.7
830.3 		−2.7
+5.0
−11.6
23 7/1 3368.826 2300 3364.997 7/4
26/15 		968.8
952.3 		−3.8
+12.7
24 189/25 3502.063 2400 3511.302 19/10
40/21
36/19
21/11
189/100 		1111.2
1115.5
1106.4
1119.5
1102.1 		+0.1
−4.2
+4.9
−8.2
+9.2
25 25/3 3670.672 2500 3657.606 33/32
25/24 		53.3
70.7 		+4.3
−13.1
26 9/1 3803.910 2600 3803.910 9/8 203.9 0
27 243/25 3937.148 2700 3950.214 11/9
27/22
243/200 		347.4
354.5
337.1 		+2.8
−4.3
+13.1
28 75/7 4105.757 2800 4096.518 4/3
75/56 		498.0
505.8 		−1.5
−9.2
29 81/7 4238.994 2900 4242.823 81/56
36/25 		639.0
631.3 		+3.8
+11.5
30 63/5 4386.422 3000 4389.127 30/19
63/40
128/81
19/12
11/7
100/63 		790.8
786.4
792.2
795.6
782.5
799.9. 		−1.7
2.7
−3.1
−6.5
+6.6
−10.8
31 675/49 4540.841 3100 4535.431 12/7
675/392
19/11 		933.1
940.8
946.2 		+2.3
−5.4
+10.8
32 15/1 4688.269 3200 4681.735 28/15
15/8
13/7 		1080.6
1088.3
1071.7 		+1.1
−6.5
+10.0
33 81/5 4821.506 3300 4828.040 64/63
81/80 		27.3
21.5 		+0.7
+6.5
34 441/25 4968.934 3400 4974.344 21/19
441/400
10/9
11/10 		173.3
168.9
182.4
165.0 		+1.0
+5.4
−8.1
+9.3
35 135/7 5123.353 3500 5120.648 135/112
6/5 		323.4
315.6 		−2.7
+5.0
36 21/1 5270.781 3600 5266.952 17/13
21/16
98/75
64/49
13/10 		464.4
470.8
463.1
462.3
454.2
 +2.5
−3.8
+3.9
+4.6
+12.7
37 567/25 5404.018 3700 5413.257 64/45
10/7
567/400 		609.8
617.5
604.0 		+3.5
−4.2
+9.2
38 25/1 5572.627 3800 5559.561 14/9
25/16 		764.9
772.6 		−5.3
−13.1
39 27/1 5705.865 3900 5705.865 27/16 905.9 0
40 729/25 5839.103 4000 5852.169 11/6
729/400 		1049.4
1039.1 		+2.8
+13.1
41 225/7 6007.712 4100 5998.473 1/1
225/224 		0.0
7.7 		−1.5
+9.2
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