The Bohlen-Pierce Site
BP Names and Definitions

This page deals with names and definitions of scales, intervals etc. as related to the Bohlen-Pierce scale. It does not provide explanations; they can be found on other, subject-related pages at the BP site.

Basic just intonation BP interval names:

 Span of interval (ratio) Name 27/25 BP first 25/21 BP second 9/7 BP third 7/5 BP fourth 75/49 BP fifth 5/3 BP sixth 9/5 BP seventh 49/25 BP eighth 15/7 BP ninth 7/3 BP tenth 63/25 BP eleventh 25/9 BP twelfth 3/1 BP thirteenth

This scale containes four different semitones:

- Small semitone 27/25
- Minor semitone 49/45
- Major semitone 375/343
- Great semitone 625/567

The differences between these semitones are defined as dieses. A simple way to imagine the relationship between semitones and dieses is the semitone diamond (not shown "to scale" in the following picture):

 Name Ratio Cent Small BP diesis (D - d) 16875/16807 = 1.0040 6.990 Minor BP diesis (d) 245/243 = 1.0082 14.191 Major BP diesis (D) 3125/3087 = 1.0123 21.181 Great BP diesis (D + d) 15625/15309 = 1.0206 35.371

Developing the JI scale from other base tones than 1/1 leads to enharmonics, arranged in a "chain of semitone diamonds":

 Diamond # Interval name Interval span [ratio] Exp's a,b,c Interval span [cent] - (reference) 1/1 0,0,0 0 I BP first 27/25 3,-2,0 133 49/45 -2,-1,2 147 375/343 1,3,-3 154 625/567 -4,4,-1 169 II BP second 729/625 6,-4,0 266 147/125 1,-3,2 281 (405/343) 4,1,-3 288 25/21 -1,2,-1 302 III BP third (3969/3125) 4,-5,2 414 2401/1875 -1,-4,4 428 9/7 2,0,-1 435 35/27 -3,1,1 449 IV BP fourth 243/175 5,-2,-1 568 7/5 0,-1,1 583 (3375/2401) 3,3,-4 590 625/441 -2,4,-2 604 V BP fifth 189/125 3,-3,1 716 343/225 -2,-2,3 730 75/49 1,2,-2 737 125/81 -4,3,0 751 VI BP sixth (5103/3125) 6,-5,1 849 1029/625 1,-4,3 863 81/49 4,0,-2 870 5/3 -1,1,0 884 VII BP seventh 9/5 2,-1,0 1018 49/27 -3,0,2 1032 625/343 0,4,-3 1039 (3125/1701) -5,5,-1 1053 VIII BP eighth 243/125 5,-3,0 1151 49/25 0,-2,2 1165 675/343 3,2,-3 1172 125/63 -2,3,-1 1186 IX BP ninth 1323/625 3,-4,2 1298 (2401/1125) -2,-3,4 1312 15/7 1,1,-1 1319 175/81 -4,2,1 1334 X BP tenth 81/35 4,-1,-1 1453 7/3 -1,0,1 1467 5625/2401 2,4,-4 1474 (3125/1323) -3,5,-2 1488 XI BP eleventh 63/25 2,-2,1 1600 (343/135) -3,-1,3 1614 125/49 0,3,-2 1621 625/243 -5,4,0 1635 XII BP twelfth 1701/625 5,-4,1 1733 343/125 0,-3,3 1748 135/49 3,1,-2 1755 25/9 -2,2,0 1769 - BP thirteenth 3/1 1,0,0 1902

The distances between diamonds are called "links".

Complete BP interval list based on a suggestion by Manuel Op de Coul:

(Nc = number of coincidences)

 Interval Class Nc Ratio Cents Name 0 - 1/1 0 unison 1 5 27/25 133.238 great limma, BP small semitone 4 49/45 147.428 BP minor semitone 2 375/343 154.418 BP major semitone 2 625/567 168.609 BP great semitone 2 1 729/625 266.475 4 147/125 280.666 0 405/343 287.656 8 25/21 301.847 BP second, quasi-tempered minor third 3 0 3969/3125 413.903 1 2401/1875 428.094 8 9/7 435.084 septimal major third, BP third 4 35/27 449.275 9/4-tone, septimal semi-diminished fourth 4 2 243/175 568.322 8 7/5 582.512 septimal tritone, BP fourth 0 3375/2401 589.503 3 625/441 603.693 5 4 189/125 715.750 2 343/225 729.940 5 75/49 736.931 BP fifth 2 125/81 751.121 6 0 5103/3125 848.987 2 1029/625 863.178 3 81/49 870.168 8 5/3 884.359 major sixth, BP sixth 7 8 9/5 1017.596 just minor seventh, BP seventh 3 49/27 1031.787 2 625/343 1038.777 0 3125/1701 1052.968 8 2 243/125 1150.834 octave minus maximal diesis 5 49/25 1165.024 BP eighth 2 675/343 1172.015 4 125/63 1186.205 9 3 1323/625 1298.262 0 2401/1125 1312.452 8 15/7 1319.443 septimal minor ninth, BP ninth 2 175/81 1333.633 10 4 81/35 1452.680 8 7/3 1466.871 minimal tenth, BP tenth 1 5625/2401 1473.861 0 3125/1323 1488.052 11 8 63/25 1600.108 BP eleventh, quasi-equal major tenth 0 343/135 1614.299 4 125/49 1621.289 1 625/243 1635.480 12 2 1701/625 1733.346 2 343/125 1747.537 4 135/49 1754.527 5 25/9 1768.717 classic augmented eleventh, BP twelfth 0 - 3/1 1901.955 just twelfth, BP thirteenth, "tritave"

Tone names in diatonic scales
(the reference scale here is the Lambda mode):

 Tone no. Relation to base tone Name (*) Distance to previous tone 1 1/1 C - 2 25/21 D 25/21 3 9/7 E 27/25 4 7/5 F 49/45 5 5/3 G 25/21 6 9/5 H 27/25 7 15/7 J 25/21 8 7/3 A 49/45 9 25/9 B 25/21 10 3/1 C' 27/25

(*) Named in accordance with a suggestion by Manuel op de Coul, aiming at making memorizing easier. The names previously used by Heinz Bohlen were: i, l, m, n, o, r, s, t, u, i'.

Note

Because of the site having entered a maintenance only status, several recent systems of notation and tone names, created by Stephen Fox, Georg Hajdu and Nora-Louise Müller, are not listed here. There may be others, too.

Notation

Reference scale (c-lambda):

Key signatures for other lambda scales:

Note

Because of the site having entered a maintenance only status, several recent systems of notation and tone names, created by Stephen Fox, Georg Hajdu and Nora-Louise Müller, are not listed here. There may be others, too.

List of at least somewhat evaluated diatonic modes
(1 indicates a semitone step, 2 a whole tone step):

1212 1 1212 "Dur I" (Bohlen, 1972)
1212 1 1221 "Gamma" (Bohlen, 1972)
2112 1 2112 "Dur II" (Bohlen, 1972)
2121 1 2121 "Moll I", later "Delta" (Bohlen, 1972)
1212 1 2121 "Moll II", later "Pierce" (Bohlen, 1972, and Pierce, independently in 1984)
1212 1 2112 "Harmonic scale" (Bohlen, 1997)
2112 1 2121 "Lambda" (Bohlen, 1997).

Diatonic modes in form of a table:

 Mode => Dur I Dur II Moll I (Delta) Moll II (Pierce) Gamma Harmonic Lambda 1/1 X X X X X X X 27/25 X X X X 25/21 X X X 9/7 X X X X X X X 7/5 X X X X X X 75/49 X 5/3 X X X X X X X 9/5 X X X X X X X 49/25 X X 15/7 X X X X X 7/3 X X X X X X X 63/25 X X X 25/9 X X X X 3/1 X X X X X X X

Commas, depending on the interval ratio chosen to "complete" the cycle:

 Step no. n Ratio R Comma C C [cent] 1 27/25 0.90645 -170 2 25/21 1.0718 120 3 9/7 0.97166 -49.8 4 7/5 0.97989 -35.2 5 75/49 1.0415 70.2 6 5/3 1.0503 84.9 7 9/5 0.95212 - 84.9 8 49/25 0.96019 -70.3 9 15/7 1.0205 35.1 10 7/3 1.0292 49.8 11 63/25 0.93297 -120 12 25/9 1.1031 170

List of interval elements and their structures

There are always three independent basic building blocks that can be used to describe all other intervals: two dieses and one semitone. Those chosen for the list below are the small diesis (a = D - d), the minor diesis (d), and the small semitone (ST).

 Name or symbol Numerical parameters m,n,o (3m 5n 7o) Structure Cents small diesis a 3, 4, - 5 a 6.990 minor diesis d - 5, 1, 2 d 14.191 major diesis D - 2, 5, - 3 a + d 21.181 great diesis - 7, 6, - 1 a + 2d 35.371 7/5-comma - 4, - 13, 13 3a + d 35.161 7/3-comma 23, 0, - 13 4d - a 49.772 49/25-comma 8, 26, - 26 6a + 2d 70.323 5/3-comma - 19, 13, 0 2a + 5d 84.933 25/21-comma - 15, 26, - 13 5a + 6d 120.095 27/25-comma 38, - 26, 0 4a +10d 169.867 small semitone 3, - 2, 0 ST 133.238 minor semitone - 2, - 1, 2 ST + d 147.428 major semitone 1, 3, - 3 ST + a + d 154.418 great semitone - 4, + 4, -1 ST + a + 2d 168.609 small link 10, - 8, 1 ST - a -2d 97.866 minor link 5, - 7, 3 ST - a - d 112.057 major link 8, - 3, - 2 ST - d 119.047 great link 3, - 2, 0 ST 133.238

Wide Triad: 3:5:7, for example CGA

Narrow Triad: 5:7:9, for example CFH

Bohlen had originally named 3:5:7 the "major triad" and 5:7:9 the "minor triad". In several discussions, especially with Paul Erlich, it turned out, however, that these names unavoidably triggered the expectation that there should exist a BP tonality fully parallel to that of the traditional Western scale. To prevent people from jumping to conclusions in this respect, "wide" and "narrow" triad have been introduced as working titles.

Pitch correlation between BP and the traditional Western scale:

 BP (JI) Traditional (ET) Pitch [Hz] A1 E 82.4 A b + 2 cent 247.2