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Bohlen–Pierce scale

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Chord from just Bohlen–Pierce scale: C-G-A, tuned to harmonics 3, 5, and 7. "BP" above the clefs indicates Bohlen–Pierce notation.[citation needed] Playⓘ
Same chord in Ben Johnston's notation for just intonation

The Bohlen–Pierce scale (BP scale) is a musical tuning and scale, first described in the 1970s, that offers an alternative to the octave-repeating scales typical in Western and other musics,[1] specifically the equal-tempered diatonic scale.

The interval 3:1 (often called by a new name, tritave) serves as the fundamental harmonic ratio, replacing the diatonic scale's 2:1 (the octave) with a perfect twelfth (an octave higher than a perfect fifth). For any pitch that is part of the BP scale, all pitches one or more tritaves higher or lower are part of the system as well, and are considered equivalent.

The BP scale divides the tritave into 13 steps, either equal tempered (the most popular form), or in a justly tuned version. Compared with octave-repeating scales, the BP scale's intervals are more consonant with certain types of acoustic spectra.[citation needed]

The scale was independently described by Heinz Bohlen,[2] Kees van Prooijen[3] and John R. Pierce. Pierce, who, with Max Mathews and others, published his discovery in 1984,[4] renamed the Pierce 3579b scale and its chromatic variant the Bohlen–Pierce scale after learning of Bohlen's earlier publication. Bohlen had proposed the same scale based on consideration of the influence of combination tones on the Gestalt impression of intervals and chords.[5]

The intervals between BP scale pitch classes are based on odd integer frequency ratios, in contrast with the intervals in diatonic scales, which employ both odd and even ratios found in the harmonic series. Specifically, the BP scale steps are based on ratios of integers whose factors are 3, 5, and 7. Thus the scale contains consonant harmonies based on the odd harmonic overtones 3:5:7:9 (playⓘ). The chord formed by the ratio 3:5:7 (playⓘ) serves much the same role as the 4:5:6 chord (a major triad playⓘ) does in diatonic scales (3:5:7 = 1:⁠1+2/3⁠:⁠2+1/3⁠ and 4:5:6 = 2:⁠2+1/2⁠:3 = 1:⁠1+1/4⁠:⁠1+1/2⁠).

Chords and modulation

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3:5:7 s intonation sensitivity pattern is similar to 4:5:6 s (the just major chord), more similar than that of the minor chord.[6] This similarity suggests that our ears will also perceive 3:5:7 as consonant.

The 3:5:7 chord may thus be considered the major triad of the BP scale. It is approximated by an interval of 6 equal-tempered BP semitones (play one semitoneⓘ) on bottom and an interval of 4 equal-tempered semitones on top (semitones 0, 6, 10; playⓘ). A minor triad is correspondingly 6 semitones on top and 4 semitones on bottom (0, 4, 10; playⓘ). 5:7:9 is the first inversion of the major triad (0, 4, 7; playⓘ).[7]

A study of chromatic triads formed from arbitrary combinations of the 13 tones of the chromatic scale among twelve musicians and twelve untrained listeners found semitones 0, 1, 2 to be the most dissonant chord (playⓘ), but 0, 11, 13 (playⓘ) was considered the most consonant by the trained subjects (because it sounds like an octave-dropped major triad) and 0, 7, 10 (playⓘ) was judged most consonant by the untrained subjects.[8]

Every tone of the Pierce 3579b scale is in a major and minor triad except for tone II of the scale. There are thirteen possible keys. Modulation is possible through changing a single note. Moving note II up one semitone causes the tonic to rise to what was note III (semitone 3), which therefore may be considered the dominant. One may consider VIII (semitone 10) the analogue of the subdominant.[7]

Timbre and the tritave

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Bohlen-Pierce clarinet family: soprano, contra(bass) and tenor

3:1 serves as the fundamental harmonic ratio, replacing the diatonic scale's 2:1 (the octave). (playⓘ) This interval is a perfect twelfth in diatonic nomenclature (perfect fifth when reduced by an octave), but as this terminology is based on step sizes and functions not used in the BP scale, it is often called by a new name, tritave (playⓘ), in BP contexts, referring to its role as a pseudooctave, and using the prefix "tri-" (three) to distinguish it from the octave. In conventional scales, if a given pitch is part of the system, then all pitches one or more octaves higher or lower also are part of the system and, furthermore, are considered equivalent. In the BP scale, if a given pitch is present, then none of the pitches one or more octaves higher or lower are present, but all pitches one or more tritaves higher or lower are part of the system and are considered equivalent.

The BP scale's use of odd integer ratios is appropriate for timbres containing only odd harmonics. Because the clarinet's spectrum (in the chalumeau register) consists of primarily the odd harmonics, and the instrument overblows at the twelfth (or tritave) rather than the octave as most other woodwind instruments do, there is a natural affinity between it and the Bohlen–Pierce scale. At the suggestion of composer Georg Hajdu, clarinet maker Stephen Fox developed the first Bohlen–Pierce soprano clarinets and began offering them for sale in early 2006. He produced the first BP tenor clarinet (six steps below the soprano) in 2010 and the first epsilon clarinet (four steps above the soprano) in 2011. A contra clarinet (one tritave lower than the soprano) is now (2020) played by Nora Mueller, Luebeck, Germany.

Just tuning

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A diatonic Bohlen–Pierce scale may be constructed with the following just ratios (chart shows the "Lambda" (λ) scale):

Note Name C D E F G H J A B C
Degree scale degree 1 scale degree 2 scale degree 3 scale degree 4 scale degree 5 scale degree 6 scale degree 7 scale degree 8 scale degree 9 scale degree 1
Ratio 1:1 25:21 9:7 7:5 5:3 9:5 15:7 7:3 25:9 3:1
Cents 0 301.85 435.08 582.51 884.36 1017.60 1319.44 1466.87 1768.72 1901.96
Midi Cⓘ Dⓘ Eⓘ Fⓘ Gⓘ Hⓘ Jⓘ Aⓘ Bⓘ Cⓘ
Step Name T s S T s T S T s
Ratio 25:21 27:25 49:45 25:21 27:25 25:21 49:45 25:21 27:25
Cents 301.85 133.24 147.43 301.85 133.24 301.85 147.43 301.85 133.24

play just Bohlen–Pierce "Lambda" scaleⓘ contrast with just major diatonic scaleⓘ

A just BP scale may be constructed from four overlapping 3:5:7 chords, for example, V, II, VI, and IV, though different chords may be chosen to produce a similar scale:[9] 

(5:3)(7:5)
V  IX  III
        |
       III VII I
               |
            VI I IV
                 |
                 IV VIII II

Bohlen–Pierce temperament

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"Chromatic circle" for the Bohlen–Pierce scale, with the third mode of the Lambda scale marked. The VII degree is mistakenly marked twice.[1]

Bohlen originally expressed the BP scale in both just intonation and equal temperament. The tempered form, which divides the tritave into thirteen equal steps, has become the most popular form. Each step is 13√3 = 31⁄13 = 1.08818… above the next, or 1200 log2 (31⁄13) = 146.3… cents per step. The octave is divided into a fractional number of steps. Twelve equally tempered steps per octave are used in 12-tet. The Bohlen–Pierce scale could be described as 8.202087-tet, because a full octave (1200 cents), divided by 146.3… cents per step, gives 8.202087 steps per octave.

Dividing the tritave into 13 equal steps tempers out, or reduces to a unison, both of the intervals 245:243 (about 14 cents, sometimes called the minor Bohlen–Pierce diesis) and 3125:3087 (about 21 cents, sometimes called the major Bohlen–Pierce diesis) in the same way that dividing the octave into 12 equal steps reduces both 81:80 (syntonic comma) and 128:125 (5-limit limma) to a unison. A 7-limit linear temperament tempers out both of these intervals; the resulting Bohlen–Pierce temperament no longer has anything to do with tritave equivalences or non-octave scales, beyond the fact that it is well adapted to using them. A tuning of 41 equal steps to the octave (1200⁄41 = 29.27 cents per step) would be quite logical for this temperament. In such a tuning, a tempered perfect twelfth (1902.4 cents, about a half cent larger than a just twelfth) is divided into 65 equal steps, resulting in a seeming paradox: Taking every fifth degree of this octave-based scale yields an excellent approximation to the non-octave-based equally tempered BP scale. Furthermore, an interval of five such steps generates (octave-based) MOSes (moments of symmetry) with 8, 9, or 17 notes, and the 8-note scale (comprising degrees 0, 5, 10, 15, 20, 25, 30, and 35 of the 41-equal scale) could be considered the octave-equivalent version of the Bohlen–Pierce scale.

Intervals and scale diagrams

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The following are the thirteen notes in the scale (cents rounded to nearest whole number):

Justly tuned

Interval (cents) 133 169 133 148 154 147 134 147 154 148 133 169 133
Note name C D♭ D E F G♭ G H J♭ J A B♭ B C
Note (cents) 0 133 302 435 583 737 884 1018 1165 1319 1467 1600 1769 1902

Equal-tempered

Interval (cents) 146 146 146 146 146 146 146 146 146 146 146 146 146
Note name C C♯/D♭ D E F F♯/G♭ G H H♯/J♭ J A A♯/B♭ B C
Note (cents) 0 146 293 439 585 732 878 1024 1170 1317 1463 1609 1756 1902

play equal tempered Bohlen–Pierce scaleⓘ

Steps Name EQ interval Cents in EQ Just intonation interval Traditional name Cents in just intonation Difference
0 C 30⁄13 = 1.00 0.00 1:1 = 1.00 Unison 0.00 0.00
1 C♯/D♭ 31⁄13 = 1.09 146.30 27:25 = 1.08 Great limma 133.24 13.06
2 D 32⁄13 = 1.18 292.61 25:21 = 1.19 Quasi-tempered minor third 301.85 −9.24
3 E 33⁄13 = 1.29 438.91 9:7 = 1.29 Septimal major third 435.08 3.83
4 F 34⁄13 = 1.40 585.22 7:5 = 1.40 Lesser septimal tritone 582.51 2.71
5 F♯/G♭ 35⁄13 = 1.53 731.52 75:49 = 1.53 BP fifth 736.93 −5.41
6 G 36⁄13 = 1.66 877.83 5:3 = 1.67 Just major sixth 884.36 −6.53
7 H 37⁄13 = 1.81 1024.13 9:5 = 1.80 Greater just minor seventh 1017.60 6.53
8 H♯/J♭ 38⁄13 = 1.97 1170.44 49:25 = 1.96 BP eighth 1165.02 5.42
9 J 39⁄13 = 2.14 1316.74 15:7 = 2.14 Septimal minor ninth 1319.44 −2.70
10 A 310⁄13 = 2.33 1463.05 7:3 = 2.33 Septimal minimal tenth 1466.87 −3.82
11 A♯/B♭ 311⁄13 = 2.53 1609.35 63:25 = 2.52 Quasi-tempered major tenth 1600.11 9.24
12 B 312⁄13 = 2.76 1755.66 25:9 = 2.78 Classic augmented eleventh 1768.72 −13.06
13 C 313⁄13 = 3.00 1901.96 3:1 = 3.00 Just twelfth, "tritave" 1901.96 0.00

Music and composition

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Octave 12-tet (left) compared with tritave 13-tet (right)

What does music using a Bohlen–Pierce scale sound like, aesthetically? Dave Benson suggests it helps to use only sounds with only odd harmonics, including clarinets or synthesized tones, but argues that because "some of the intervals sound a bit like intervals in [the more familiar] twelve-tone scale, but badly out of tune", the average listener will continually feel "that something isn't quite right", due to social conditioning.[10]

Mathews and Pierce conclude that clear and memorable melodies may be composed in the BP scale, that "counterpoint sounds all right", and that "chordal passages sound like harmony", presumably meaning progression, "but without any great tension or sense of resolution".[11] In their 1989 study of consonance judgment, both intervals of the five chords rated most consonant by trained musicians are approximately diatonic intervals, suggesting that their training influenced their selection and that similar experience with the BP scale would similarly influence their choices.[8]

Compositions using the Bohlen–Pierce scale include "Purity", the first movement of Curtis Roads' Clang-Tint.[12] Other computer composers to use the BP scale include Jon Appleton, Richard Boulanger (Solemn Song for Evening (1990)), Georg Hajdu, Juan Reyes' ppP (1999-2000),[13] Ami Radunskaya's "A Wild and Reckless Place" (1990),[14] Charles Carpenter (Frog à la Pêche (1994) & Splat),[15][16] and Elaine Walker (Stick Men (1991), Love Song, and Greater Good (2011)).[17]

David Lieberman, an associate professor of architecture at the University of Toronto, directed the development of a "Stredici", a string instrument tuned to the Bohlen–Pierce scale. The five-meter long instrument was used in concerts in Boston in 2012.[18]

Symposium

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A first Bohlen–Pierce symposium took place in Boston on March 7 to 9, 2010, produced by composer Georg Hajdu (Hochschule für Musik und Theater Hamburg) and the Boston Microtonal Society. Co-organizers were the Boston Goethe Institute, the Berklee College of Music, the Northeastern University and the New England Conservatory of Music. The symposium participants, which included Heinz Bohlen, Max Mathews, Clarence Barlow, Curtis Roads, David Wessel, Psyche Loui, Richard Boulanger, Georg Hajdu, Paul Erlich, Ron Sword, Julia Werntz, Larry Polansky, Manfred Stahnke, Stephen Fox, Elaine Walker, Todd Harrop, Gayle Young, Johannes Kretz, Arturo Grolimund, Kevin Foster, presented 20 papers on history and properties of the Bohlen–Pierce scale, performed more than 40 compositions in the novel system and introduced several new musical instruments. Performers included German musicians Nora-Louise Müller and Ákos Hoffman on Bohlen–Pierce clarinets and Arturo Grolimund on Bohlen–Pierce pan flute as well as Canadian ensemble tranSpectra, and US American xenharmonic band ZIA, led by Elaine Walker.

Other unusual tunings or scales

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Other non-octave tunings investigated by Bohlen[19] include twelve steps in the tritave, named A12 by Enrique Moreno[20] and based on the 4:7:10 chord 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 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.[21] The Bohlen 833 cents scale is based on the Fibonacci sequence, although it was created from combination tones, 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".[22]

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).[23] This gives the alpha scale 15.39 steps per octave and the beta scale 18.75 steps per octave.[24]

Expansions

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39-tone equal division of the tritave

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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.[25]

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
38 2.9167 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
36 2.7569 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
33 2.5335 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
31 2.3947 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
29 2.2635 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
20 1.7566 975.36 135/77 972.03 3.33
7/4 968.83 6.54
19 1.7078 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
10 1.3254 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
8 1.2528 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
6 1.1841 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
3 1.0882 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
1 1.0286 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

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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-tone equal division of the octave except that each step is slightly smaller (less than a hundredth of a cent per step).

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
60 2.7569 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
23 1.4751 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
14 1.2670 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
5 1.0882 146.30 12/11 150.6371 -4.33
49/45 147.4281 -1.12
4 1.0699 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
2 1.0344 58.52 28/27 62.9609 -4.44
33/32 53.2729 5.25
1 1.0170 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

See also

[edit]

References

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  1. ^ a b Pierce, John R. (2001). "Consonance and scales". In Cook, Perry R (ed.). Music, Cognition, and Computerized Sound: An Introduction to Psychoacoustics. MIT Press. p. 183. ISBN 978-0-262-53190-0.
  2. ^ Bohlen, Heinz (1978). "13 Tonstufen in der Duodezime". Acoustica (in German). 39 (2). Stuttgart: S. Hirzel Verlag: 76–86. Retrieved 27 November 2012.
  3. ^ Prooijen, Kees van (1978). "A Theory of Equal-Tempered Scales". Interface. 7: 45–56. doi:10.1080/09298217808570248. Retrieved 27 November 2012.
  4. ^ Mathews, M.V.; Roberts, L.A.; Pierce, J.R. (1984). "Four new scales based on nonsuccessive-integer-ratio chords". J. Acoust. Soc. Am. 75, S10(A) (S1): S10. Bibcode:1984ASAJ...75...10M. doi:10.1121/1.2021272.
  5. ^ Mathews, Max V.; Pierce, John R. (1989). "The Bohlen–Pierce Scale". In Mathews, Max V.; Pierce, John R. (eds.). Current Directions in Computer Music Research. MIT Press. p. 167. ISBN 9780262631396.
  6. ^ Mathews & Pierce (1989), pp. 165–166
  7. ^ a b Mathews & Pierce (1989), pp. 169
  8. ^ a b Mathews & Pierce (1989), pp. 171
  9. ^ Mathews; Pierce (1989). p. 170.
  10. ^ Benson, Dave. "Musical scales and the Baker's Dozen". Musik og Matematik. 28/06: 16.
  11. ^ Mathews; Pierce (1989). p. 172.
  12. ^ Thrall, Michael Voyne (Summer 1997). "Synthèse 96: The 26th International Festival of Electroacoustic Music". Computer Music Journal. 21 (2): 90–92 [91]. doi:10.2307/3681110. JSTOR 3681110.
  13. ^ "John Pierce (1910-2002)". Computer Music Journal. 26, No. 4 (Languages and Environments for Computer Music): 6–7. Winter 2002.
  14. ^ Discography of Microtonal CDs, Huygens-Fokker Foundation, retrieved 2016-12-13.
  15. ^ d'Escrivan, Julio (2007). Collins, Nick (ed.). The Cambridge Companion to Electronic Music. Cambridge Companions to Music. p. 229. ISBN 9780521868617.
  16. ^ Benson, Dave (2006). Music: A Mathematical Offering. p. 237. ISBN 9780521853873.
  17. ^ "Concerts". Bohlen-Pierce-Conference.org. Retrieved 27 November 2012.
  18. ^ "David Lieberman participates in the Electric Fields and Mini-Maker Faire | Daniels". www.daniels.utoronto.ca. Retrieved 2023-01-04.
  19. ^ Bohlen (1978). footnote 26, page 84.
  20. ^ "Other Unusual Scales". The Bohlen–Pierce Site. Retrieved 27 November 2012. Cites: Moreno, Enrique Ignacio (Dec 1995). "Embedding Equal Pitch Spaces and The Question of Expanded Chromas: An Experimental Approach". Dissertation. Stanford University: 12–22.
  21. ^ "Other Unusual Scales", The Bohlen–Pierce Site. Retrieved 27 November 2012. Cites: Bohlen (1978). pp. 76–86.
  22. ^ Bohlen, Heinz. "An 833 Cents Scale". The Bohlen–Pierce Site. Retrieved 27 November 2012.
  23. ^ Sethares, William (2004). Tuning, Timbre, Spectrum, Scale. p. 60. ISBN 1-85233-797-4.
  24. ^ Carlos, Wendy (2000) [1986]. "Liner notes". Beauty in the Beast (CD). Wendy Carlos. ESD. 81552.
  25. ^ "BP Scale Structures". The Bohlen–Pierce Site. Retrieved 27 November 2012.
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