Intervals of BP: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
No edit summary
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
Reorganized table, emphasized standard JI values, added hekts, increased precision to 3 digits (except for the nearby JI columns), fixed a few errors, added a few lines at the bottom to reach ~5 octaves
Line 1: Line 1:
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, which also known as [[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"
! rowspan="2" |degree
! colspan="2" |Just intonation
! colspan="2" |Equal-tempered
! colspan="3" |Nearby JI intervals, [[Octave reduction|octave-reduced]]
|-
|-
! | degree
!(ratio)
! | cents value
!([[Cent|¢]])
! | nearby frequency ratios
!([[Hekt|hekts]])
! | cents value
!(¢)
! | difference between BP interval & just interval
! | (ratio)
|-
! | ([[Cent|¢]])
! colspan="3" | first octave and first tritave...
! | error in 13edt (¢)
| |  
| |
|-
|-
| | 0
| | 0
|1/1
|0
|0
| | 0
| | 0
| | 1/1
| | 1/1
Line 20: Line 27:
|-
|-
| | 1
| | 1
| | 146.3
|27/25
|133.238
|100
| | 146.304
| | 12/11
| | 12/11


13/12
13/12


''27/25''
'''27/25'''
| | 150.6
| | 150.6


138.6
138.6


''133.2''
'''133.2'''
| | -4.3
| | -4.3


7.7
+7.7


''13.1''
'''+13.1'''
|-
|-
| | 2
| | 2
| | 292.6
|25/21
|301.847
|200
| | 292.608
| | 32/27
| | 32/27


Line 45: Line 58:
19/16
19/16


25/21
'''25/21'''
| | 294.1
| | 294.1


Line 52: Line 65:
297.5
297.5


301.8
'''301.8'''
| | -1.5
| | -1.5


3.4
+3.4


-4.9
-4.9


-9.2
'''-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
| | 585.217
| | '''7/5'''


45/32
45/32
| | 582.5
| | '''582.5'''


590.2
590.2
| | 2.7
| | '''+2.7'''


-5.0
-5.0
|-
|-
| | 5
| | 5
| | 731.5
|75/49
|736.931
|500
| | 731.521
| | 32/21
| | 32/21


26/17
26/17


75/49
'''75/49'''


49/32
49/32


''20/13''
20/13
| | 729.2
| | 729.2


735.6
735.6


736.9
'''736.9'''


737.7
737.7


''745.8''
745.8
| | 2.3
| | +2.3


-4.1
-4.1


-5.4
'''-5.4'''


-6.2
-6.2


''-14.3''
-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
|1017.596
|700
| | 1024.130
| | 38/21
| | 38/21


29/16
29/16


9/5
'''9/5'''


20/11
20/11
Line 128: Line 156:
1029.6
1029.6


1017.6
'''1017.6'''


1035.0
1035.0
| | -2.6
| | -2.6


-5.5
-5.5


6.5
'''+6.5'''


-10.9
-10.9
|-
|-
| | 8
| | 8
| | 1170.4
|49/25
| | 49/25
|1165.024
| | 1165.0
|800
| | 5.4
| | 1170.434
|-
| | '''49/25'''
! colspan="3" | second octave...
| | '''1165.0'''
| |
| | '''+5.4'''
| |
|-
|-
| | 9
| | 9
| | 116.7
|15/7
| | 15/14
|1319.443
|900
| | 1316.738
| | '''15/14'''


16/15
16/15


14/13
14/13
| | 119.4
| | '''119.4'''


111.7
111.7


128.3
128.3
| | -2.7
| | '''-2.7'''


5.0
+5.0


-11.6
-11.6
|-
|-
| | 10
| | 10
| | 263.0
|7/3
| | 7/6
|1466.871
|1000
| | 1463.042
| | '''7/6'''


22/19
22/19
| | 266.9
| | '''266.9'''


253.8
253.8
| | -3.9
| | '''-3.8'''


9.2
+9.2
|-
|-
| | 11
| | 11
| | 409.3
|63/25
|1600.108
|1100
| | 1609.347
| | 19/15
| | 19/15


Line 189: Line 225:
14/11
14/11


63/50
'''63/50'''
| | 409.2
| | 409.2


Line 198: Line 234:
417.5
417.5


400.1
'''400.1'''
| | .1
| | +0.1


1.5
+1.5


4.9
+4.9


-8.2
-8.2


9.2
'''+9.2'''
|-
|-
| | 12
| | 12
| | 555.7
|25/9
|1768.717
|1200
| | 1755.651
| | 11/8
| | 11/8


Line 217: Line 256:
26/19
26/19


25/18
'''25/18'''
| | 551.3
| | 551.3


Line 224: Line 263:
543.0
543.0


568.7
'''568.7'''
| | 4.4
| | +4.4


-7.7
-7.7


12.7
+12.7


-13.0
'''-13.1'''
|-
! colspan="3" | second tritave...
| |
| |
|-
|-
| | 13
| | 13
| | 702.0
|3/1
| | 3/2
|1901.955
| | 702.0
|1300
| | 0
| | 1901.955
| | '''3/2'''
| | '''702.0'''
| | '''0'''
|-
|-
| | 14
| | 14
| | 848.3
|81/25
|2035.193
|1400
| | 2048.259
| | 18/11
| | 18/11


13/8
13/8
'''81/50'''
| | 852.6
| | 852.6


840.5
840.5
| | -4.3


7.8
'''835.2'''
| |  -4.3
 
+7.8
 
'''+13.1'''
|-
|-
| | 15
| | 15
| | 994.6
|25/7
|2203.802
|1500
| | 2194.563
| | 16/9
| | 16/9
'''25/14'''
| | 996.1
| | 996.1
| | -1.5
'''1003.8'''
| | -1.5
'''-9.2'''
|-
|-
| | 16
| | 16
| | 1140.9
|27/7
| | 27/14
|2337.039
| | 1137.0
|1600
| | 3.9
| | 2340.868
|-
| | '''27/14'''
! colspan="3" | third octave...
| | '''1137.0'''
| |
| | '''+3.8'''
| |
|-
|-
| | 17
| | 17
| | 87.2
|21/5
|2484.467
|1700
| | 2487.172
| | 20/19
| | 20/19


21/20
'''21/20'''


19/18
19/18
Line 282: Line 337:
| | 88.8
| | 88.8


84.5
'''84.5'''


93.6
93.6


80.5
80.5
| | -1.6
| | -1.6


2.7
'''+2.7'''


-6.4
-6.4


6.7
+6.7
|-
|-
| | 18
| | 18
| | 233.5
|225/49
|2638.886
|1800
| | 2633.476
| | 8/7
| | 8/7
'''225/196'''


15/13
15/13
| | 231.2
| | 231.2


''247.7''
'''238.9'''
| | 2.3
 
247.7
| | +2.3


''-14.2''
'''-5.4'''
 
-14.2
|-
|-
| | 19
| | 19
| | 379.8
|5/1
| | 5/4
|2786.314
|1900
| | 2779.780
| | '''5/4'''


26/21
26/21
| | 386.3
| | '''386.3'''


369.7
369.7
| | -6.5
| | '''-6.5'''


10.1
+10.1
|-
|-
| | 20
| | 20
| | 526.1
|27/5
|2919.551
|2000
| | 2926.085
| | 19/14
| | 19/14


27/20
'''27/20'''


15/11
15/11
| | 528.7
| | 528.7


519.6
'''519.6'''


537.0
537.0
| | -2.6
| | -2.6


6.5
'''+6.5'''


-10.9
-10.9
|-
|-
| | 21
| | 21
| | 672.4
|147/25
|3066.979
|2100
| | 3072.389
| | 28/19
| | 28/19


25/17
25/17
'''147/100'''


22/15
22/15
Line 347: Line 421:


667.7
667.7
'''667.0'''


663.0
663.0
| | 1.1
| | +1.1


4.7
+4.7


9.4
'''+5.4'''
 
+9.4
|-
|-
| | 22
| | 22
| | 818.7
|45/7
| | 8/5
|3221.398
|2200
| | 3218.693
| | '''45/28'''
8/5


21/13
21/13
| | 813.7
| | '''821.4'''
813.7


830.3
830.3
| | 5.0
| | '''-2.7'''
+5.0


-11.6
-11.6
|-
|-
| | 23
| | 23
| | 965.0
|7/1
| | 7/4
|3368.826
|2300
| | 3364.997
| | '''7/4'''


26/15
26/15
| | 968.8
| | '''968.8'''


952.3
952.3
| | -3.8
| | '''-3.8'''


12.7
+12.7
|-
|-
| | 24
| | 24
| | 1111.3
|189/25
|3502.063
|2400
| | 3511.302
| | 19/10
| | 19/10


Line 388: Line 478:


21/11
21/11
'''189/100'''
| | 1111.2
| | 1111.2


Line 395: Line 487:


1119.5
1119.5
| | 0.1
 
'''1102.1'''
| | +0.1


-4.2
-4.2


4.9
+4.9


-8.2
-8.2
|-
 
! colspan="3" | fourth octave...
'''+9.2'''
| |
| |
|-
|-
| | 25
| | 25
| | 57.6
|25/3
|3670.672
|2500
| | 3657.606
| | 33/32
| | 33/32
'''25/24'''
| | 53.3
| | 53.3
| | 4.3
'''70.7'''
|-
| | +4.3
! colspan="3" | third tritave...
'''-13.1'''
| |
| |
|-
|-
| | 26
| | 26
| | 203.9
|9/1
| | 9/8
|3803.910
| | 203.9
|2600
| | 0
| | 3803.910
| | '''9/8'''
| | '''203.9'''
| | '''0'''
|-
|-
| | 27
| | 27
| | 350.2
|243/25
|3937.148
|2700
| | 3950.214
| | 11/9
| | 11/9


27/22
27/22
'''243/200'''
| |347.4
| |347.4
354.55
354.5
| | +2.8
 
-3.35
337.1
| | +2.8
-4.3
 
'''+13.1'''
|-
|-
| | 28
| | 28
| | 496.5
|75/7
|4105.757
|2800
| | 4096.518
| | 4/3
| | 4/3
| | 498.05
'''75/56'''
| | +2.05
| | 498.0
'''505.8'''
| | -1.5
'''-9.2'''
|-
|-
| | 29
| | 29
| | 642.8
|81/7
| |36/25
|4238.994
| |631.3
|2900
| | +11.3.
| | 4242.823
| |'''81/56'''
36/25
| |'''639.0'''
631.3
| | '''+3.8'''
+11.5
|-
|-
| | 30
| | 30
| | 789.1
|63/5
|4386.422
|3000
| | 4389.127
| |30/19
| |30/19
'''63/40'''
128/81
128/81


Line 456: Line 578:
100/63
100/63
| |790.8
| |790.8
'''786.4'''
792.2
792.2


Line 463: Line 586:


799.9.
799.9.
| | -1.7
| | -1.7
'''2.7'''


-3.1
-3.1
Line 474: Line 598:
|-
|-
| | 31
| | 31
| | 935.4
|675/49
|4540.841
|3100
| | 4535.431
| |12/7
| |12/7
'''675/392'''
19/11
19/11
| |933.1
| |933.1
'''940.8'''
946.2
946.2
| | +2.3
| | +2.3
'''-5.4'''
+10.8
+10.8
|-
|-
| | 32
| | 32
| | 1081.7
|15/1
| |15/8
|4688.269
28/15
|3200
 
| | 4681.735
| |28/15
'''15/8'''
13/7
13/7
| |1088.3
| |1080.6
1080.6
'''1088.3'''
 
1071.7
1071.7
| | -6.6
| | +1.1
 
'''-6.5'''
+1.1


+10.0
+10.0
|-
| colspan="3" style="text-align:center;" | '''fifth octave...'''
| |
| |
|-
|-
| | 33
| | 33
| | 28.0
|81/5
|4821.506
|3300
| | 4828.040
| |64/63
| |64/63
'''81/80'''
| |27.3
| |27.3
| | +0.7
'''21.5'''
| | +0.7
'''+6.5'''
|-
|-
| |34
| |34
| |174.3
|441/25
| |11/10
|4968.934
21/19
|3400
| |4974.344
| |21/19
'''441/400'''


10/9
10/9
| |165.0
 
173.3
11/10
| |173.3
'''168.9'''


182.4
182.4
| | +9.3
 
+1.0
165.0
| | +1.0
'''+5.4'''


-8.1
-8.1
+9.3
|-
|-
| |35
| |35
| |320.65
|135/7
| |6/5
|5123.353
| |315.6
|3500
| | +5.05
| |5120.648
| |'''135/112'''
6/5
| |'''323.4'''
315.6
| | '''-2.7'''
+5.0
|-
|-
| |36
| |36
| |466.95
|21/1
| |21/16
|5270.781
|3600
| |5266.952
| |'''21/16'''
17/13
17/13


Line 539: Line 689:


13/10
13/10
| |470.8
| |'''470.8'''
464.4
464.4


Line 548: Line 698:
454.2
454.2


| | -3.85
| | '''-3.8'''
+2.55
+2.5


+3.85
+3.9


+4.65
+4.6


+12.75
+12.7
|-
|-
| |37
| |37
| |613.3
|567/25
|5404.018
|3700
| |5413.257
| |10/7
| |10/7
64/45
64/45
567/400
| |617.5
| |617.5
609.8
609.8
| | -4.2
 
'''604.0'''
| | -4.2
+3.5
+3.5
'''+9.2'''
|-
|-
| |38
| |38
| |759.6
|25/1
|5572.627
|3800
| |5559.561
| |14/9
| |14/9
'''25/16'''
| |764.9
| |764.9
| | -5.3
'''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'''
|}
|}

Revision as of 19:12, 21 August 2021

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, which also known as 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) (¢) (hekts) (¢) (ratio) (¢) 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 21/16

17/13

98/75

64/49

13/10

470.8

464.4

463.1

462.3

454.2

-3.8

+2.5

+3.9

+4.6

+12.7

37 567/25 5404.018 3700 5413.257 10/7

64/45

567/400

617.5

609.8

604.0

-4.2

+3.5

+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

Retrieved from "https://en.xen.wiki/index.php?title=Intervals_of_BP&oldid=76682"