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NickKNIqpa:

[[File:BP chord 357 just.png|thumb|right|Chord from just Bohlen–Pierce scale: C-G-A, tuned to harmonics 3, 5, and 7. "BP" above the clefs indicates Bohlen–Pierce notation. {{Audio|BP Just 357 chord.ogg|Play}}]]

The '''Bohlen–Pierce scale''' ('''BP scale''') is a musical [[Scale (music)|scale]] that offers an alternative to the [[octave]]-repeating scales typical in [[Classical music|Western]] and other musics, specifically the [[diatonic scale]].<ref>{{cite book | title = Music, Cognition, and Computerized Sound: An Introduction to Psychoacoustics | author = John R. Pierce | chapter = Consonance and scales | editor = Perry R. Cook | publisher = MIT Press | year = 2001 | isbn = 978-0-262-53190-0 | page = 183 | url = http://books.google.com/books?id=L04W8ADtpQ4C&pg=PA183&dq=%22Bohlen-Pierce+scale%22+13+octave&lr=&as_brr=0&as_pt=ALLTYPES&ei=2jhdSYDPMYnwkQSi1LXSAw }}</ref> Compared with octave-repeating scales, its [[interval (music)|interval]]s are more [[consonance|consonant]] with certain types of acoustic [[frequency spectrum|spectra]]. It was independently described by Heinz Bohlen,<ref>
http://www.huygens-fokker.org/bpsite/publication0178.html H. Bohlen, "13 Tonstufen in der Duodezime," ''Acoustica'' 39, 76-86 (1978).</ref> Kees van Prooijen<ref>
http://www.kees.cc/tuning/interface.html K. van Prooijen, "A Theory of Equal-Tempered Scales," ''Interface'' 7, 45-56 (1978).</ref> and [[John R. Pierce]]. Pierce, who, with [[Max Mathews]] and others, published his discovery in 1984,<ref>
M.V. Mathews, L.A. Roberts, and J.R. Pierce, "Four new scales based on nonsuccessive-integer-ratio chords," ''J. Acoust. Soc. Amer.'' 75, S10(A) (1984).</ref> 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 tone]]s on the [[Gestalt psychology|Gestalt]] impression of intervals and chords.<ref name="Current Directions, p.167">
Max V. Mathews and John R. Pierce (1989). "The Bohlen–Pierce Scale", p.167. ''Current Directions in Computer Music Research'', Max V. Mathews and John R. Pierce, eds. MIT Press.</ref>

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 (music)|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 ({{Audio|3579 Harmonic Chord.ogg|play}}). The chord formed by the ratio 3:5:7 ({{Audio|BP Just 357 chord.ogg|play}}) serves much the same role as the 4:5:6 chord (a major triad {{Audio|JI 456 chord.ogg|play}}) does in diatonic scales (3:5:7 = 1:1.66:2.33 and 4:5:6 = 2:2.5:3 = 1:1.25:1.5).

==Chords and modulation==
3:5:7's [[Intonation (music)#Intonation sensitivity|intonation sensitivity]] pattern is similar to 4:5:6's (the just major chord), more similar than that of the minor chord.<ref name="Current Directions, p.165-66">
Mathews and Pierce (1989). "The Bohlen–Pierce Scale", p.165-66.</ref> This similarity suggests that our ears will also perceive 3:5:7 as harmonic.

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 [[semitone]]s ({{Audio|BP ET half step.ogg|play one semitone}}) on bottom and an interval of 4 equal-tempered semitones on top (semitones: 0,6,10; {{Audio|BP ET 357.ogg|play}}). A minor triad is thus 6 semitones on top and 4 semitones on bottom (0,4,10; {{Audio|BP ET minor.ogg|play}}). 5:7:9 is the first inversion of the major triad (0,4,7; {{Audio|BP ET 579.ogg|play}}).<ref name="Current Directions, p.169">Mathews and Pierce (1989). "The Bohlen–Pierce Scale", p.169.</ref>

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 0,1,2 (semitones) to be the most dissonant chord ({{Audio|BP 012.ogg|play}}) but 0,11,13 ({{Audio|BP 0 11 13.ogg|play}}) was considered the most consonant by the trained subjects and 0,7,10 ({{Audio|BP 0 7 10.ogg|play}}) was judged most consonant by the untrained subjects.<ref name="Current Directions, p.171">
Mathews and Pierce (1989). "The Bohlen–Pierce Scale", p.171.</ref>

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 may be considered the [[dominant (music)|dominant]]. VIII (semitone: 10) may be considered the [[subdominant]].<ref name="Current Directions, p.169"/>

==Timbre and the tritave==
3:1 serves as the fundamental harmonic ratio, replacing the diatonic scale's 2:1 (the [[octave]]). ({{Audio|Octave.ogg|play}}) This interval is a perfect twelfth in [[diatonic scale|diatonic]] nomenclature ([[perfect fifth]] when reduced by an octave), but as this terminology is based on step sizes and [[diatonic function|functions]] not used in the BP scale, it is often called by a new name, '''''tritave''''' ({{Audio|Tritave.ogg|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 [[octave equivalency|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. In early 2006 clarinet maker [[Stephen Fox (clarinet maker)|Stephen Fox]] began offering Bohlen–Pierce soprano clarinets for sale, and 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, while a contra clarinet (one tritave lower than the soprano) is under development.

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

{| class="wikitable" style="text-align:center"
!
| colspan="2" | '''C'''
| colspan="2" | '''D'''
| colspan="2" | '''E'''
| colspan="2" | '''F'''
| colspan="2" | '''G'''
| colspan="2" | '''H'''
| colspan="2" | '''J'''
| colspan="2" | '''A'''
| colspan="2" | '''B'''
| colspan="2" | '''C'''
|-
! Ratio
| colspan="2" | 1/1
| colspan="2" | [[Semitone maximus|25/21]]
| colspan="2" | [[Septimal major third|9/7]]
| colspan="2" | [[Tritone|7/5]]
| colspan="2" | [[Major sixth|5/3]]
| colspan="2" | [[Minor seventh|9/5]]
| colspan="2" | [[Septimal diatonic semitone|15/7]]
| colspan="2" | [[Septimal minor third|7/3]]
| colspan="2" | [[Just chromatic semitone|25/9]]
| colspan="2" | [[Tritave|3/1]]
|-
! Step
|
| colspan="2" | T
| colspan="2" | s
| colspan="2" | s
| colspan="2" | T
| colspan="2" | s
| colspan="2" | T
| colspan="2" | s
| colspan="2" | T
| colspan="2" | s
|
|-
! Midi
| colspan="2" | {{Audio|BP Just C.ogg|C}}
| colspan="2" | {{Audio|BP Just D.ogg|D}}
| colspan="2" | {{Audio|BP Just E.ogg|E}}
| colspan="2" | {{Audio|BP Just F.ogg|F}}
| colspan="2" | {{Audio|BP Just G.ogg|G}}
| colspan="2" | {{Audio|BP Just H.ogg|H}}
| colspan="2" | {{Audio|BP Just J.ogg|J}}
| colspan="2" | {{Audio|BP Just A.ogg|A}}
| colspan="2" | {{Audio|BP Just B.ogg|B}}
| colspan="2" | {{Audio|BP Just High C.ogg|C}}
|}

{{Audio|BP Just Lambda Scale.ogg|play just Bohlen–Pierce "Lambda" scale}}
{{Audio|JI diatonic scale.ogg|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<ref name="Current Directions, p.170">Mathews and Pierce (1989). "The Bohlen–Pierce Scale", p.170.</ref>:
(5/3) (7/5)
V IX III
|
III VII I
|
VI I IV
|
IV VIII II

==Bohlen–Pierce temperament==
Bohlen originally expressed the BP scale in both [[just intonation]] and [[equal temperament]]. The [[Musical temperament|tempered]] form, which divides the tritave into thirteen equal steps, has become the most popular form. Each step is <math>3^{1/13} = 1.08818...</math> above the next, or <math>1200\log_2( 3^{1/13} )= 146.3...</math> cents per step. The octave is divided into a fractional number of steps. Twelve equally tempered steps per octave are used in [[equal temperament|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 [[regular temperament|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 temperament|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 [[cent (music)|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) [[Generated collection|MOS]]es 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==
The following are the thirteen notes in the scale (cents rounded to nearest whole number):

'''Justly tuned'''
{| class="wikitable"
|align=center bgcolor="#ffeeee"|'''Interval (cents)'''
|
|colspan=2 align=center bgcolor="#ffeeee"|133
|colspan=2 align=center bgcolor="#ffeeee"|169
|colspan=2 align=center bgcolor="#ffeeee"|133
|colspan=2 align=center bgcolor="#ffeeee"|148
|colspan=2 align=center bgcolor="#ffeeee"|154
|colspan=2 align=center bgcolor="#ffeeee"|147
|colspan=2 align=center bgcolor="#ffeeee"|134
|colspan=2 align=center bgcolor="#ffeeee"|147
|colspan=2 align=center bgcolor="#ffeeee"|154
|colspan=2 align=center bgcolor="#ffeeee"|148
|colspan=2 align=center bgcolor="#ffeeee"|133
|colspan=2 align=center bgcolor="#ffeeee"|169
|colspan=2 align=center bgcolor="#ffeeee"|133
|
|-
|align=center bgcolor="#fffbee"|'''Note name'''
|colspan=2 align=center bgcolor="#fffbee"|C
|colspan=2 align=center bgcolor="#fffbee"|D♭
|colspan=2 align=center bgcolor="#fffbee"|D
|colspan=2 align=center bgcolor="#fffbee"|E
|colspan=2 align=center bgcolor="#fffbee"|F
|colspan=2 align=center bgcolor="#fffbee"|G♭
|colspan=2 align=center bgcolor="#fffbee"|G
|colspan=2 align=center bgcolor="#fffbee"|H
|colspan=2 align=center bgcolor="#fffbee"|J♭
|colspan=2 align=center bgcolor="#fffbee"|J
|colspan=2 align=center bgcolor="#fffbee"|A
|colspan=2 align=center bgcolor="#fffbee"|B♭
|colspan=2 align=center bgcolor="#fffbee"|B
|colspan=2 align=center bgcolor="#fffbee"|C
|-
|align=center bgcolor="#eeeeff"|'''Note (cents)'''
|colspan=2 align=center bgcolor="#eeeeff"|  0  
|colspan=2 align=center bgcolor="#eeeeff"| 133 
|colspan=2 align=center bgcolor="#eeeeff"|302
|colspan=2 align=center bgcolor="#eeeeff"|435
|colspan=2 align=center bgcolor="#eeeeff"|583
|colspan=2 align=center bgcolor="#eeeeff"|737
|colspan=2 align=center bgcolor="#eeeeff"|884
|colspan=2 align=center bgcolor="#eeeeff"|1018
|colspan=2 align=center bgcolor="#eeeeff"|1165
|colspan=2 align=center bgcolor="#eeeeff"|1319
|colspan=2 align=center bgcolor="#eeeeff"|1467
|colspan=2 align=center bgcolor="#eeeeff"|1600
|colspan=2 align=center bgcolor="#eeeeff"|1769
|colspan=2 align=center bgcolor="#eeeeff"|1902
|}

'''Equal-tempered'''
{| class="wikitable"
|align=center bgcolor="#ffeeee"|'''Interval (cents)'''
|
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|colspan=2 align=center bgcolor="#ffeeee"|146
|
|-
|align=center bgcolor="#fffbee"|'''Note name'''
|colspan=2 align=center bgcolor="#fffbee"|C
|colspan=2 align=center bgcolor="#fffbee"|D♭
|colspan=2 align=center bgcolor="#fffbee"|D
|colspan=2 align=center bgcolor="#fffbee"|E
|colspan=2 align=center bgcolor="#fffbee"|F
|colspan=2 align=center bgcolor="#fffbee"|G♭
|colspan=2 align=center bgcolor="#fffbee"|G
|colspan=2 align=center bgcolor="#fffbee"|H
|colspan=2 align=center bgcolor="#fffbee"|J♭
|colspan=2 align=center bgcolor="#fffbee"|J
|colspan=2 align=center bgcolor="#fffbee"|A
|colspan=2 align=center bgcolor="#fffbee"|B♭
|colspan=2 align=center bgcolor="#fffbee"|B
|colspan=2 align=center bgcolor="#fffbee"|C
|-
|align=center bgcolor="#eeeeff"|'''Note (cents)'''
|colspan=2 align=center bgcolor="#eeeeff"|  0  
|colspan=2 align=center bgcolor="#eeeeff"| 146 
|colspan=2 align=center bgcolor="#eeeeff"|293
|colspan=2 align=center bgcolor="#eeeeff"|439
|colspan=2 align=center bgcolor="#eeeeff"|585
|colspan=2 align=center bgcolor="#eeeeff"|732
|colspan=2 align=center bgcolor="#eeeeff"|878
|colspan=2 align=center bgcolor="#eeeeff"|1024
|colspan=2 align=center bgcolor="#eeeeff"|1170
|colspan=2 align=center bgcolor="#eeeeff"|1317
|colspan=2 align=center bgcolor="#eeeeff"|1463
|colspan=2 align=center bgcolor="#eeeeff"|1609
|colspan=2 align=center bgcolor="#eeeeff"|1756
|colspan=2 align=center bgcolor="#eeeeff"|1902
|}

{{Audio|BP ET lambda scale.ogg|play equal tempered Bohlen–Pierce scale}}

{| frame="box" rules="all" cellpadding="4" style="text-align:center" align=center
|- bgcolor=#DDDDFF
!Steps
!EQ interval
!Cents in EQ
!Just intonation interval
!Traditional name
!Cents in just intonation
!Difference
|-
|0
|<math>3^\frac{0}{13}</math> = 1.00
| 0.00
|<math>\begin{matrix} \frac{1}{1} \end{matrix}</math> = 1.00
| Unison
| 0.00
| 0.00
|-
|1
|<math>3^\frac{1}{13}</math> = 1.09
| 146.30
|<math>\begin{matrix} \frac{27}{25} \end{matrix}</math> = 1.08
| Great limma
| 133.24
| 13.06
|-
|2
|<math>3^\frac{2}{13}</math> = 1.18
| 292.61
|<math>\begin{matrix} \frac{25}{21} \end{matrix}</math> = 1.19
| Quasi-tempered minor third
| 301.85
| -9.24
|-
|3
|<math>3^\frac{3}{13}</math> = 1.29
| 438.91
|<math>\begin{matrix} \frac{9}{7} \end{matrix}</math> = 1.29
| Septimal major third
| 435.08
| 3.83
|-
|4
|<math>3^\frac{4}{13}</math> = 1.40
| 585.22
|<math>\begin{matrix} \frac{7}{5} \end{matrix}</math> = 1.4
| Lesser septimal tritone
| 582.51
| 2.71
|-
|5
|<math>3^\frac{5}{13}</math> = 1.53
| 731.52
|<math>\begin{matrix} \frac{75}{49} \end{matrix}</math> = 1.53
| BP fifth
| 736.93
| -5.41
|-
|6
|<math>3^\frac{6}{13}</math> = 1.66
| 877.83
|<math>\begin{matrix} \frac{5}{3} \end{matrix}</math> = 1.67
| Just major sixth
| 884.36
| -6.53
|-
|7
|<math>3^\frac{7}{13}</math> = 1.81
| 1024.13
|<math>\begin{matrix} \frac{9}{5} \end{matrix}</math> = 1.8
| Greater just minor seventh
| 1017.60
| 6.53
|-
|8
|<math>3^\frac{8}{13}</math> = 1.97
| 1170.44
|<math>\begin{matrix} \frac{49}{25} \end{matrix}</math> = 1.96
| BP eighth
| 1165.02
| 5.42
|-
|9
|<math>3^\frac{9}{13}</math> = 2.14
| 1316.74
|<math>\begin{matrix} \frac{15}{7} \end{matrix}</math> = 2.14
| Septimal minor ninth
| 1319.44
| -2.70
|-
|10
|<math>3^\frac{10}{13}</math> = 2.33
| 1463.05
|<math>\begin{matrix} \frac{7}{3} \end{matrix}</math> = 2.33
| Septimal minimal tenth
| 1466.87
| -3.82
|-
|11
|<math>3^\frac{11}{13}</math> = 2.53
| 1609.35
|<math>\begin{matrix} \frac{63}{25} \end{matrix}</math> = 2.52
| Quasi-tempered major tenth
| 1600.11
| 9.24
|-
|12
|<math>3^\frac{12}{13}</math> = 2.76
| 1755.66
|<math>\begin{matrix} \frac{25}{9} \end{matrix}</math> = 2.78
| Classic augmented eleventh
| 1768.72
| -13.06
|-
|13
|<math>3^\frac{13}{13}</math> = 3.00
| 1901.96
|<math>\begin{matrix} \frac{3}{1} \end{matrix}</math> = 3.00
| Just twelfth, "Tritave"
| 1901.96
| 0.00
|}

==Music and composition==
What does music using a Bohlen–Pierce scale sound like, [[aesthetics of music|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] [[chromatic scale|twelve-tone scale]], but badly [[Musical tuning#Tuning practice|out of tune]]," the average listener will continually feel "that something isn't quite right," due to [[social conditioning]].<ref>
Benson, Dave. "Musical scales and the Baker’s Dozen", p.16, ''Musik og Matematik'' 28/06.</ref>

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 [[chord progression|progression]], "but without any great tension or sense of resolution."<ref name="Current Directions, p.172">
Mathews and Pierce (1989). "The Bohlen–Pierce Scale", p.172.</ref> 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.<ref name="Current Directions, p.171"/>

Compositions using the Bohlen–Pierce scale include "Purity", the first movement of [[Curtis Roads]]' ''Clang-Tint''.<ref>
"Synthèse 96: The 26th International Festival of Electroacoustic Music", p.91. Michael Voyne Thrall. ''Computer Music Journal'', Vol. 21, No. 2 (Summer, 1997), pp. 90-92.</ref> Other computer composers to use the BP scale include [[Jon Appleton]], Richard Boulanger (''Solemn Song for Evening'' (1990)), [[Georg Hajdu]], and Juan Reyes' ''[http://ccrma.stanford.edu/~juanig/descrips/ppPdesc.html ppP]'' (1999-2000).<ref>"John Pierce (1910-2002)". ''Computer Music Journal'', Vol. 26, No. 4, Languages and Environments for Computer Music (Winter, 2002), pp. 6-7.</ref> Also Charles Carpenter (''Frog à la Pêche'' (1994) & ''Splat'').<ref>d'Escrivan, Julio (2007). ''The Cambridge Companion to Electronic Music'', p.229. Nick Collins, ed. ISBN 9780521868617.</ref><ref>Benson, Dave (2006). ''Music: A Mathematical Offering'', p.237. ISBN 9780521853873.</ref>

==Symposium==
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 Institut]]e, 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.

==Other unusual tunings or scales==
Other non-octave tunings investigated by Bohlen include twelve steps in the tritave, named A12 by Enrique Moreno <ref>
Moreno, Enrique Ignacio: Embedding Equal Pitch Spaces and The Question of Expanded Chromas: An Experimental Approach. Dissertation, Stanford University, Dec. 1995, pp. 12 - 22. Cited in [http://www.huygens-fokker.org/bpsite/otherscales.html "Other Unusual Scales"], ''The Bohlen–Pierce Site''.</ref> and based on the 4:7:10 chord {{audio|A12 4 7 10 on C.mid|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 {{audio|5 7 9 chord on E.mid|Play}} and of which only the just version would be used.<ref>
Bohlen, Heinz: 13 Tonstufen in der Duodezime. Acustica, vol.39 no. 2, S. Hirzel Verlag, Stuttgart, 1978, pp. 76 - 86. Cited in [http://www.huygens-fokker.org/bpsite/otherscales.html "Other Unusual Scales"], ''The Bohlen–Pierce Site''.</ref> The Bohlen 833 cents scale is based on the [[Fibonacci sequence]], although it was created from [[combination tone]]s, 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".<ref>
http://www.huygens-fokker.org/bpsite/833cent.html "An 833 Cents Scale", ''The Bohlen–Pierce Site''.</ref>

An expansion of the Bohlen–Pierce tritave from 13 equal steps to 39 equal steps, proposed by Paul Erlich, 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>
http://www.huygens-fokker.org/bpsite/scales.html "BP Scale Structures", ''The Bohlen–Pierce Site''.</ref>

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).<ref>Sethares, William (2004). ''Tuning, Timbre, Spectrum, Scale'', p.60. ISBN 1-85233-797-4.</ref> This gives the alpha scale 15.39 steps per octave and the beta scale 18.75 steps per octave.<ref>Carlos, Wendy (2000/1986). "Liner notes", ''Beauty in the Beast''. ESD 81552.</ref>

See also: [[Delta scale]], [[Gamma scale]].

==Footnotes==
{{reflist|2}}

==External links==
* [http://www.ziaspace.com/elaine/BP/ Bohlen–Pierce Scale Research by Elaine Walker]
* [http://www.sfoxclarinets.com/BP_sale.htm Bohlen–Pierce clarinets by Stephen Fox]
* [http://www.huygens-fokker.org/bpsite/ The Bohlen–Pierce Site: Web place of an alternative harmonic scale]
* [http://www.kees.cc/music/scale13/scale13.html Kees van Prooijen's BP page]
* [http://www.ziaspace.com/ZIA/mp3s/LoveSong_BP_EW.mp3 song in Bohlen Pierce Scale]
* [http://bohlen-pierce-conference.org/ Bohlen–Pierce symposium]
{{scales}}
{{musical tuning}}

{{DEFAULTSORT:Bohlen–Pierce Scale}}
[[Category:Microtonality]]
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Main Page

2014-08-10T20:49:18Z

NickKNIqpa:

{{Distinguish|Ambiophonics}}
'''Ambisonics''' is a series of recording and replay techniques using multichannel [[Audio mixing (recorded_music)|mixing]] technology that can be used live or in the studio. By encoding and decoding sound information on a number of channels, a 2-dimensional ("planar", or horizontal-only) or 3-dimensional ("periphonic<ref>Michael A. Gerzon, ''Periphony: With-Height Sound Reproduction''. Journal of the Audio Engineering Society, 1973, 21(1):2–10.</ref>", or full-sphere) sound field can be presented. Ambisonics was invented by [[Michael Gerzon]] of the [[Mathematical Institute|Mathematical Institute, Oxford]], who – with Professor Peter Fellgett<ref>Peter Fellgett, ''Ambisonics. Part One: General System Description'', Studio Sound, August 1975, 1:20–22,40.</ref> of the [[University of Reading]], David Brown, John Wright and John Hayes of the now defunct IMF Electronics<ref>{{cite web | url =http://www.imf-electronics.com/Home/imf/ambisonic | title =Ambisonic | work =Home of the Transmission Line Loudspeakers | publisher =IMF Electronics | accessdate =9 March 2012}}</ref>, and building on the work of other researchers – developed the theoretical and practical aspects of the system in the early 1970s.

== Advantages ==

Ambisonics offers a number of advantages over other [[surround sound]] systems:
* It is [[isotropic]] in that sounds arriving from all directions are treated equally (as opposed to most other surround systems that assume that the main sources of sound are frontal and that rear channels are only for ambience or special effects).
* All speakers are generally used to localise a sound in any direction (as opposed to conventional pan-potted (pair-wise mixing) techniques which use only two adjacent speakers). This gives better localisation, particularly to the sides and rear.<ref>{{cite journal |last=Gerzon |first=Michael |authorlink=Michael Gerzon |date=8 December 1977 |title=Don't say quad – say psychoacoustics |journal=New Scientist |volume=76 |pages=634–636 }}</ref><ref>{{cite web |url=http://members.tripod.com/martin_leese/Ambisonic/experiment.html#REFERENCES |title=References on Pair-wise Mixing |accessdate=24 January 2007 |last=Leese |first=Martin |date=6 February 2005 |work=An Experiment into Pair-Wise Mixing and Channel Separation }}</ref>
* The stability and imaging of the reproduced soundfield vary less with listener position than with most other surround systems. The soundfield can even be appreciated by listeners ''outside'' the speaker array.<ref>{{cite journal |author=Malham, DG |year=1992 |title=Experience with Large Area 3-D Ambisonic Sound Systems |journal=Proceedings of the Institute of Acoustics |volume=14 |issue=5 |pages=209–215 |url=http://www.dmalham.freeserve.co.uk/ioapaper1.pdf |format=PDF |accessdate=24 January 2007 }}</ref>
* A minimum of four channels of information are required for distribution and storage of a full-sphere soundfield, and three for a horizontal soundfield. (This is fewer than other surround systems). Full-sphere replay requires a minimum of six loudspeakers (a minimum of four for horizontal), the signal for each speaker position being derived using appropriate circuitry or software.
* The loudspeakers do not have to be positioned in a rigid setting; most regular polygons and (with somewhat more complex technology) a number of irregular figures can be accommodated. This allows the speaker configuration to be matched more closely to real listening environments, such as domestic living rooms.
* The Ambisonic signal is independent of the replay-system: the same signal can be decoded for varying numbers of loudspeakers (in general, the more speakers, the higher the accuracy of the reconstructed soundfield). This allows flexibility for composers, performers and production teams to produce a "final" mix without worrying about how the mix will later be released and decoded.

== Disadvantages ==

Ambisonics also suffers from some disadvantages:
* It is not supported by any major record label or media company.
* It has never been well marketed and, largely as a result, is not widely known.
* It can be conceptually difficult for people to grasp (as opposed to conventional "one channel=one speaker" surround, which is easier).
* It requires an Ambisonic decoder box at the replay end, and there are few commercial decoder manufacturers. However, [[#G-Format|G-Format]] ameliorates this (with attendant benefits and drawbacks), and there is a growing collection of free Ambisonic software decoders.
* The minimum number of loudspeakers required for planar (horizontal) decoding is four. While this is satisfactory in the average sized living-room for which it was designed, if the listening area is too large then, without treatment, the resulting soundfield can approach the limits of stability. This has resulted in some unimpressive demos. A six-speaker horizontal array is more stable.
* The two-channel matrixed form of Ambisonics, 2-channel [[#UHJ format|UHJ]], is not comparable to "true multichannel" (discrete) surround distribution systems. While multichannel distribution formats for Ambisonics exist (such as B-Format, G-Format and 2½ to 4 channel UHJ), only 2-channel UHJ and, to a lesser extent, G-Format have been employed in commercial releases to date.

== First-order Ambisonics and B-Format ==

In the basic version, known as ''first-order Ambisonics,'' sound information is encoded into four channels: ''W'', ''X'', ''Y'' and ''Z''. This is called Ambisonic B-format. The ''W'' channel is the non-directional mono component of the signal, corresponding to the output of an omnidirectional microphone. The ''X'', ''Y'' and ''Z'' channels are the directional components in three [[dimension]]s. They correspond to the outputs of three figure-of-eight microphones, facing forward, to the left, and upward respectively. (Note that the fact that B-format channels are analogous to microphone configurations does ''not'' mean that Ambisonic recordings can only be made with coincident microphone arrays.)

The B-format signals are based on a [[spherical harmonic]] decomposition of the soundfield and correspond to the [[Sound#Sound_pressure|sound pressure]] (''W''), and the three components of the pressure gradient (''X'', ''Y'', and ''Z'') (not to be confused with the related [[particle velocity]]) at a point in space. Together, these approximate the sound field on a sphere around the microphone; formally the first-order truncation of the [[multipole expansion]]. This is called "first-order" because ''W'' (the mono signal) is the zero-order information, corresponding to a sphere (constant function on the sphere), while ''X,'' ''Y,'' and ''Z'' are the first-order terms (the dipoles), corresponding to the response of figure-of-eight microphones – as functions, to particular functions that are positive on half the sphere, and negative of the other half. This first-order truncation is only an approximation of the overall sound field (but see [[#Higher-order Ambisonics|Higher-order Ambisonics]]).

The [[loudspeaker]] signals are derived by using a [[linear combination]] of these four channels, where each signal is dependent on the actual position of the speaker in relation to the center of an imaginary sphere the surface of which passes through all available speakers. In more advanced decoding schemes, spatial equalization is applied to the signals to account for the differences in the high- and low-frequency [[sound localization]] mechanisms in human hearing. A further refinement accounts for the distance of the listener from the loudspeakers.

=== Decoding ===
{{Details|Ambisonic decoding}}
Several different decoder designs are possible, with different advantages and disadvantages. They use different decoding equations, and are intended for different types of application. Hardware decoders have been commercially available since the late 1970s; currently, Ambisonics is standard in surround products offered by [[Meridian Audio, Ltd.]]. Ad hoc software decoders are also available (see [[#Downloadable B-Format files|Downloadable B-Format files]]).

=== Relationship to coincident stereo techniques ===
Different linear combinations of ''W'', ''X'', ''Y'' and ''Z'' can create signals equivalent to those picked up by ''any'' conventional [[microphone]] (omnidirectional, cardioid, hypercardioid, etc.) pointing in ''any'' direction. Thus the signals used in any coincident stereo microphone technique can be generated directly from the B-format signals (for example, Blumlein Mid-Side with a forward-facing cardioid using <math>M = \sqrt{2} W + X\,\!</math> and <math>S = Y\,\!</math>, or a [[Blumlein Pair]] using <math>L = (X + Y) / \sqrt{2}</math> and <math>R = (X - Y) / \sqrt{2}</math>).


Thus we can consider first-order B Format as a series of sum and
difference channels:
* ''W'' = front + back + left + right + up + down (mono, omni mic)
* ''X'' = front − back (figure-of-eight mic facing forward)
* ''Y'' = left − right (figure-of-eight facing left)
* and ''Z'' = up − down (figure-of-eight facing up).

== Downloadable B-Format files ==


An official file format for B-Format files, called
[http://members.tripod.com/martin_leese/Ambisonic/B-Format_file_format.html ".amb" format],
has been defined. Over two hundred such files are available for free download from [http://www.ambisonia.com/ Ambisonia.com]. The website also gives details of
[http://www.ambisonia.com/wiki/index.php/Playback_Software software players].

The ".amb" file format is defined for B-Format files up to third-order, full-sphere (16 channels), although most of the files currently available are first-order, full-sphere (4 channels).

== Recording techniques ==

See also.<ref>Michael A. Gerzon, ''Ambisonics. Part Two: Studio Techniques'', Studio Sound, October 1975, pages 24–30. Correction in Oct. 1975 issue on page 60.</ref>

=== The soundfield microphone ===
Many Ambisonic recordings have been made using a special microphone – the [[soundfield microphone]] (SFM). This microphone has also become popular with recording engineers, since it can be reconfigured electronically or via software to provide different stereo and 3-D polar responses either during or after recording.

=== "Native" microphones ===
The SFM uses a tetrahedral array of capsules, the outputs of which are matrixed together to generate the component B-Format signals. However it is entirely practical to generate B-Format from a collection of coincident microphones (or mic capsules), each with the characteristics of one of the B-Format channels listed earlier. This is referred to as a "Native" Ambisonic microphone or microphone array. The primary difficulty inherent in this approach is that high-frequency localisation relies on the diaphragms approaching true coincidence, and this is difficult to achieve with complete microphones. However electronic coincidence compensation can be used, and this can be effective especially where small capsules and not whole microphones are employed.

Thus if you wish to generate planar B-Format (WXY), you could use an omnidirectional mic coincident with a forward-facing and a left-facing figure-of-eight. Exactly this technique was used by Dr Jonathan Halliday at [[Nimbus Records]] to record their extensive and continuing series of Ambisonic releases.

=== Ambisonic mixing ===
A popular and unfortunate misconception is that Ambisonic recordings can only be made with the SFM, and as a result there is a widespread, and erroneous, belief that Ambisonics can only be used to capture a live acoustic event (something that accounts for a tiny proportion of modern commercial recordings, the vast majority of which are built up in the studio and mixed from multitrack). This is not the case. In fact, [[Michael Gerzon]]'s designs for Ambisonic panpots pre-date much of his work on soundfield microphone technology. Ambisonic panpots – which allow mono (for example) signals to be localised in B-Format space – were developed as early as the 1970s, and were incorporated into a special mixing console designed by Chris Daubney<ref>Chris Daubney, ''Ambisonics – an operational insight''. Studio Sound, Aug. 1982, pp.52–58</ref> at the IBA (UK [[Independent Broadcasting Authority]]) and built by Alice Stancoil Ltd in the early 1980s for the IBA surround-sound test broadcasts.

Ambisonic panpots, with differing degrees of sophistication, provide the fundamental additional studio tool required to create an Ambisonic mix, by making it possible to localise individual, conventionally-recorded multi-track or multi-mic sources around a 360° stage analogous to the way conventional stereo panpots localise sounds across a front stage. However, unlike stereo panpots, which traditionally vary only the level between two channels, Ambisonic panning provides additional cues which eliminate conventional localisation accuracy problems. This is especially pertinent to surround, where our ability to localise level-only panned sources is severely limited to the sides and rear.

Other tools included "spreaders" which were designed to "de-localise" a signal (typically by varying the virtual source angle with frequency within a determined range) – for example, in the case of reverb returns – however these were not developed further.

==== Legacy hardware ====
[[Image:Adr.jpg|frame|right|Audio & Design's Ambisonic Mastering System. From top to bottom, the B-Format Converter, the UHJ Transcoder, the Ambisonic Decoder, and the Pan-Rotate unit.]]By the early 1980s, studio hardware existed for the creation of multitrack-sourced, Ambisonically-mixed content, including the ability to incorporate SFM-derived sources (for example for room ambience) into a multichannel mix.<ref>Richard Elen, [http://www.ambisonic.net/ambimix.html ''Ambisonic mixing – an introduction''], Studio Sound, September 1983</ref> This was thanks primarily to the efforts of Dr Geoffrey Barton (now of Trifield Productions) and the pro-audio manufacturers Audio & Design Recording, UK (now Audio & Design Reading Ltd). Barton designed a suite of outboard rack-mounted studio units that became known as the Ambisonic Mastering System.<ref>Michael A Gerzon and Geoffrey J. Barton, ''Ambisonic Surround-Sound Mixing for Multitrack Studios'', AES Preprint C1009, Convention 2i (April 1984)(AES E-Library location: (CD aes10) /pp8185/pp8405/9109.pdf)</ref> These units were patched into a conventional mixing console and allowed conventional multitrack recordings to be mixed Ambisonically. The system consisted of four units:
* Pan-Rotate Unit – This enabled eight mono signals to be panned in B-format, including 360° "angle" control and a "radius vector" control allowing the source to be brought in towards the centre, plus a control to rotate an external or internal B-format signal.
* B-Format Converter – This connected to four groups and an aux send and allowed existing console panpots to pan across a B-Format quadrant.
* UHJ Transcoder – This both encoded B-Format into 2-channel UHJ (see [[Ambisonic UHJ Format|UHJ Format]]) and in addition allowed a stereo front stage and a stereo rear stage (both with adjustable widths) to be transcoded direct to 2-channel UHJ.
* Ambisonic Decoder – this accepted both horizontal (WXY) B-format and 2-channel UHJ and decoded it to four speaker feeds with configurable array geometry.

It is understood that versions of these units were subsequently made available in the late 1990s by Cepiar Ltd along with some other Ambisonics hardware. It is not known if they are still currently available.

A significant number of releases were made with this equipment, all in 2-channel UHJ, including several albums on the KPM production music library label, and commercial releases such as Steve Hackett's ''Till We Have Faces'', The Alan Parsons Project's ''Stereotomy'', Paul McCartney's ''Liverpool Oratorio'', Frank Perry's ''Zodiac'', a series of albums on the Collins Classics label, and others, most of which are available on CD. See ''The Ambisonic Discography'' in the [[#External links|External links]] for more information. Engineer John Timperley employed a transcoder on virtually all his mixes over the course of over a dozen years until his death in 2006. Unfortunately the albums, film soundtracks and other projects he created in UHJ over this period are largely undocumented at present, and thus remain unlisted in the Discography.

The lack of availability of 4-track mastering equipment led to a tendency (now regretted by some of the people involved) to mix directly to 2-channel UHJ rather than recording B-format and then converting it to UHJ for release. The fact that you could mix direct to 2-channel UHJ with nothing more than the transcoder made this even more tempting. As a result there is a lack of legacy Ambisonically-mixed B-format recordings that could be released today in more advanced formats (such as G-Format). However, the remastering – and in some cases release – of original 2-channel UHJ recordings in G-Format has proved to be surprisingly effective, yielding results at least as good as the original studio playbacks, thanks primarily to the significantly higher quality of current decoding systems (such as file-based software decoders) compared to those available when the recordings were made.

==== Current mixing tools ====
The advent of digital audio workstations has led to the development of both encoding and decoding tools for Ambisonic production. Many of these have been developed under the auspices of the University of York (see [[#External links|External links]]). The vast majority to date have been created using the VST plugin standard developed by Steinberg and used widely in a number of commercial and other software-based audio production systems, notably Steinberg's Nuendo. With the lack of necessity to interface to a conventional console, the encoding tools have primarily taken the form of B-Format panpots and associated controls. Decoder plugins are available
for monitoring.

There are presently some issues with implementing B-format groups and other channel structures in current DAW software which is often either stereo-based or based inflexibly on conventional surround configurations. The ability must exist to use plugins with one input and multiple outputs, for example, and it must be possible to create B-format buses of some sort and hook up decoder plugins to them, record their contents, and perform other operations. Documentation is being assembled to assist engineers wishing to work with these tools.

There are also stand-alone software tools for manipulating multichannel files and for offline decoding of B-Format and UHJ files to standard arrays, plus software players capable of playing and decoding standard B-Format files and other Ambisonic content.

The plugin field is a particular growth area for Ambisonic production tools at the present time.

== UHJ format ==

{{Details|Ambisonic UHJ format}}
UHJ is a development of Ambisonics designed to allow Ambisonic recordings to be carried by mono- and stereo-compatible media. It is a hierarchy of systems in which the recorded soundfield will be reproduced with a degree of accuracy that varies according to the available channels. Although UHJ permits the use of up to four channels (carrying full-sphere with-height surround), only the 2-channel variant is in current use (as it is compatible with currently-available 2-channel media). 2-channel UHJ does not include height information and decodes to provide a horizontal surround experience to a somewhat lower level of resolution than 2½- or 3-channel UHJ.

== Super stereo ==

A feature of domestic Ambisonic decoders has been the inclusion of a ''super stereo'' feature. This allows conventional stereo signals to be "wrapped around" the listener, using some of the capabilities of the decoder. A control is provided that allows the width to be varied between mono-like and full surround. This provides a useful capability for a listener to get more from their existing stereo collection.

A different kind of "super stereo" is experienced by listeners to a 2-channel UHJ signal who are not using a decoder. Because of the inter-channel phase relationships inherent in the encoding scheme, the listener experiences stereo that is often significantly wider than the loudspeakers. It is also often more stable and offers superior imaging.

Both features were used as selling points in the early days of Ambisonics, and especially Ambisonic mixing. It helped to overcome a "chicken and egg" situation where record companies were reluctant to release Ambisonic recordings because there were few decoders in the marketplace, while hi-fi manufacturers were unwilling to licence and incorporate Ambisonic decoders in their equipment because there was not very much mainstream released content. On the one hand, it was worth having a decoder because you could get more out of your existing record collection; while on the other it was worth making Ambisonic recordings because even people without a decoder could gain appreciable benefits.

== G-Format ==

The lack of availability of Ambisonic decoders (only a handful of hardware decoder models are currently available, although software-based players are now emerging) led to the proposal that Ambisonics could be distributed by decoding the original signal (preferably B-Format but also legacy 2-channel UHJ recordings) ''in the studio'' instead of at the listening end. A professional software or hardware-based decoder is used to decode the Ambisonic signal to a conventional surround speaker array (e.g. 5.1) and the resulting speaker feeds are authored to a conventional multichannel disc medium such as DVD. This is known as "G-Format".<ref>Richard Elen, [http://www.ambisonic.net/gformat.html ''Ambisonics for the New Millennium''], September 1998.</ref>

The obvious advantage of this approach is that any surround listener can be able to experience Ambisonics; no special decoder is required beyond that found in a common home theatre system. The main disadvantage is that the flexibility of rendering a single, standard Ambisonic signal to any target speaker array is lost: the signal is targeted towards a specific "standard" array and anyone listening with a different array may experience a degradation of localisation accuracy, depending on how much the actual array differs from the target.

In practice, Ambisonics in general has proved to be very robust, however. Examples of G-Format recently released by [[Nimbus Records]] used 2-channel UHJ decoded to a square array of four speakers (this is conventional for decoding planar Ambisonic recordings; a rectangle of sides with ratios of between 2:1 and 1:2 can be used, a square being midway between the two). The resulting 4-channel (LF, RF, LS, RS) signal was authored to DVD-Audio/Video discs and although many listeners will be listening on arrays other than a square, the results have proved very encouraging.

Some releases of G-format sourced from B-Format have also occurred, for example the album ''Swing Live'' by [[Bucky Pizzarelli]] (available on [[Chesky Records]], DVD-A or SACD), where a B-Format SFM recording was "manually decoded" to 4.0 speaker feeds in the mixdown process.

=== Recovering B-Format from G-Format ===
It is theoretically possible to recover B-Format from a G-Format signal, in which case Ambisonic listeners with their own decoders could recover the B-Format and decode it for their own array, thus achieving more accurate localisation. However for the greatest accuracy in smaller environments such as a living room, the decode process includes shelf filtering that may cause the decode to be irreversible if the shelf-filters are non-linear. It should be possible to implement linear shelf-filtering when decoding to a rectangular or regular polygonal array, but more work has yet to be performed in this area.

It is also possible that as a result of current development work (primarily by Dr Peter Craven) on hierarchical systems for audio rendering, these problems can be overcome (and G-Format superseded) by distributing a common signal that plays back as 5.1 on 5.1 systems (and so on) but can also be decoded Ambisonically if listeners have the right equipment.<ref>{{cite conference
| first = Peter G.
| last = Craven
| coauthors = Malcome J. Law, J. Robert Stuart, Rhonda J. Wilson
| year = 2003
| month = June
| title = Hierarchical Lossless Transmission of Surround Sound Using MLP
| conference = AES 24th International Conference
| conferenceurl = http://www.banffcentre.ca/aes/
| booktitle = Proceedings of the AES 24th International Conference: Multichannel Audio, The New Reality
| editor =
| others =
| edition =
| publisher = [[Audio Engineering Society|AES]]
| url =
| id =
}}</ref>

=== G-Format with height ===
It is entirely possible to create G-Format recordings that include height information. However, while there are "standards" for conventional planar surround (5.1, 7.1, etc.) there is currently no recognised standard (apart from Ambisonics) for the inclusion of height. There are several techniques being used, the most common one being to take one or two channels of the 5.1 signal (typically LFE, or CF & LFE) and to use them to drive elevated loudspeaker(s). It would be possible to decode an Ambisonic full-sphere recording to configurations like this, and to release the result (which would then be G-Format).

== Current developments ==
=== General ===
The Ogg [[Vorbis]] project has shown interest in implementing Ambisonics as a means for including surround sound in their project. In addition there is a growing series of freely-available developments such as [[Virtual Studio Technology|VST]] plugins, enabling common [[Digital audio workstation|DAW]] systems (such as [[Steinberg Nuendo|Nuendo]]) to be used to encode and decode B-Format and generate decoded speaker feeds; see [[#External links|External links]].

=== Higher-order Ambisonics ===
A particularly active area of current research is the development of "higher orders" of Ambisonics. These use more channels than the original first-order B-Format to capture significantly more spatial information. At present, "real" recording techniques using them are in their infancy, it is, however, straightforward to compose synthetic recordings. Benefits include greater localisation accuracy and better performance in large-scale replay environments such as performance spaces.

The higher orders correspond to further terms of the [[multipole expansion]] of a function on the sphere in terms of spherical harmonics. As discussed at [[wave field synthesis]], in the absence of obstacles, sound in a space over time can be described as the pressure at a plane or over a sphere – and thus if one reproduces this function, one can reproduce the sound of a microphone at any point in the space pointing in any direction.

==== Possible combinations ====
The following table lists the various higher-order combinations which are possible. In theory, the table could be extended to infinity.

In the table, note that as you move from horizontal to full-sphere, or from lower to higher orders, backwards compatibility is always guaranteed because channels are only ever added. This means, for example, that a first-order, horizontal decoder can still decode a third-order, full-sphere soundfield by simply ignoring 13 of the 16 channels.

{| class="wikitable" style="text-align:center"
|+Higher-order B-Format channels
|-
!Horizontal order
!Height order
!Soundfield type
!Number of channels
!Channels
|-
| 1|| 0||horizontal|| 3||WXY
|-
| 1|| 1||full-sphere|| 4||WXYZ
|-
| 2|| 0||horizontal|| 5||WXYUV
|-
| 2|| 1||mixed-order|| 6||WXYZUV
|-
| 2|| 2||full-sphere|| 9||WXYZRSTUV
|-
| 3|| 0||horizontal|| 7||WXYUVPQ
|-
| 3|| 1||mixed-order|| 8||WXYZUVPQ
|-
| 3|| 2||mixed-order|| 11||WXYZRSTUVPQ
|-
| 3|| 3||full-sphere|| 16||WXYZRSTUVKLMNOPQ
|}

==== Microphones and decoders ====
[[Soundfield microphone]]s for recording first-order B-Format have been commercially available for many decades. A mic which can record up to third-order is shipping.<ref>{{cite web
|url = http://www.mhacoustics.com/mh_acoustics/Eigenmike_microphone_array.html
|title = em32 Eigenmike microphone array|accessdate =18 October 2008|quote = We are currently shipping em32 arrays with spatial harmonic orders up to and including third-order.| archiveurl= http://web.archive.org/web/20081026043326/http://www.mhacoustics.com/page/page/2949006.htm| archivedate= 26 October 2008 | deadurl= no}}</ref> First-order B-Format decoders have been commercially available since the late 1970s. Ad hoc second-order and third-order software players (decoders) are currently available (see [[#Downloadable B-Format files|Downloadable B-Format files]]).

==== Use in gaming ====
Higher-order Ambisonics has found a niche market in video games developed by [[Codemasters]]. Their first game to use an Ambisonic audio engine was [[Colin McRae: DiRT]], however, this only used Ambisonics on the [[PlayStation 3]] platform.<ref>{{cite web
| url = http://etiennedeleflie.net/2007/08/30/interview-with-simon-goodwin-of-codemasters-on-the-ps3-game-dirt-and-ambisonics/
| title = Interview with Simon Goodwin of Codemasters on the PS3 game DiRT and Ambisonics.
| first = Etienne
| last = Deleflie
| date = 30 August 2007
| work = Building Ambisonia.com
| publisher = Etienne Deleflie
| location = Australia
| accessdate =7 August 2010
}}</ref> Their game [[Race Driver: GRID]] extended the use of Ambisonics to the [[Xbox 360]] platform,<ref>{{cite web
| url = http://etiennedeleflie.net/2008/06/24/codemasters-ups-their-useage-of-ambisonics-on-race-driver-grid/
| title = Codemasters ups Ambisonics again on Race Driver GRID …
| first = Etienne
| last = Deleflie
| date = 24 June 2008
| work = Building Ambisonia.com
| publisher = Etienne Deleflie
| location = Australia
| accessdate =7 August 2010
}}</ref> and [[Colin McRae: DiRT 2]] uses Ambisonics on all platforms including the PC.<ref>{{cite news
| title = Interview: Simon N Goodwin, Codemasters
| first = Ben
| last = Firshman
| url = http://theboar.org/games/2010/mar/3/interview-simon-goodwin-codemasters/
| newspaper = The Boar
| publisher = The University of Warwick
| location = Coventry, United Kingdom
| id = Core of Volume 32, Issue 11
| date = 3 March 2010
| page = 18
| accessdate =7 August 2010
}}</ref> The recent game from Codemasters, [[F1 2010 (video game)|F1 2010]], uses fourth-order Ambisonics on faster PCs. The PC versions use [[Blue Ripple Sound]]'s [[Rapture3D]] [[OpenAL]] driver.



{| class="wikitable" align="right" style="margin-left:10px;text-align:center"
|+Furse-Malham coefficients
|-
!B-Format channel!!Weight
|-
| W|| <math>1 / \sqrt{2}\,\!</math>
|-
| X|| <math>1\,\!</math>
|-
| Y|| <math>1\,\!</math>
|-
| Z|| <math>1\,\!</math>
|-
| R|| <math>1\,\!</math>
|-
| S|| <math>2 / \sqrt{3}\,\!</math>
|-
| T|| <math>2 / \sqrt{3}\,\!</math>
|-
| U|| <math>2 / \sqrt{3}\,\!</math>
|-
| V|| <math>2 / \sqrt{3}\,\!</math>
|-
| K|| <math>1\,\!</math>
|-
| L|| <math>\sqrt{45 / 32}\,\!</math>
|-
| M|| <math>\sqrt{45 / 32}\,\!</math>
|-
| N|| <math>3 / \sqrt{5}\,\!</math>
|-
| O|| <math>3 / \sqrt{5}\,\!</math>
|-
| P|| <math>\sqrt{8 / 5}\,\!</math>
|-
| Q|| <math>\sqrt{8 / 5}\,\!</math>
|}

==== Furse-Malham higher-order format ====
''Furse-Malham higher-order format'' (FMH-Format) is a set of coefficients that can be applied to the first 16 B-format channels. The FMH set of coefficients applies weightings to the channels such that all the spherical harmonic coefficients have a maximum value of unity. Whilst this approach is not rigorously "correct" in mathematical terms, it has significant engineering advantages in that it restricts the maximum levels a panned mono source will generate in some of the higher-order channels.<ref>{{cite web|url = http://www.york.ac.uk/inst/mustech/3d_audio/higher_order_ambisonics.pdf|title = Higher order Ambisonic systems|accessdate =2 November 2007|last = Malham|first = David
|year = 2003|month = April|format = PDF|work = Space in Music – Music in Space (Mphil thesis)|publisher = University of York|pages = 2–3}}</ref>

The Furse-Malham set of weighting factors is part of the ".amb" specification for [[#Downloadable B-Format files|downloadable B-Format files]].

== Patents and Trademarks ==

Most of the patents covering Ambisonic developments have now expired (including those covering the [[Soundfield microphone]]) and, as a result, the basic technology is available for anyone to implement. Exceptions to this include Dr Geoffrey Barton's [[Trifield]] technology, which is a three-speaker stereo rendering system based on Ambisonic theory ({{Cite patent|US|5594800}}), and so-called "Vienna" decoders, based on Gerzon and Barton's Vienna 1992 AES paper, which are intended for decoding to irregular speaker arrays ({{Cite patent|US|5757927}}).

The "pool" of patents comprising Ambisonics technology was originally assembled by the UK Government's National Research & Development Corporation (NRDC), which existed until the late 1970s to develop and promote British inventions and license them to commercial manufacturers – ideally to a single licensee. The system was ultimately licensed to [[Nimbus Records]] (now owned by Wyastone Estate Ltd) who hold the rights to the "interlocking circles" Ambisonic logo (UK trademarks
[http://www.patent.gov.uk/tm/t-find/t-find-number?detailsrequested=C&trademark=1113276 1113276] and
[http://www.patent.gov.uk/tm/t-find/t-find-number?detailsrequested=C&trademark=1113277 1113277]), and the text marks "AMBISONIC" and "A M B I S O N" (UK trademarks [http://www.patent.gov.uk/tm/t-find/t-find-number?detailsrequested=C&trademark=1500177 1500177] and
[http://www.patent.gov.uk/tm/t-find/t-find-number?detailsrequested=C&trademark=1112259 1112259]).

Note that applications to register the word marks (trademarks) "AMBISONICS" and "AMBISONIC" in the USA were abandoned in 1992 and 2009 (US trademark serial numbers 74118119 and 77695983). 

== Notes on nomenclature ==
=== Some terms: their meanings and usage ===
[[Michael Gerzon]] used to wryly comment on the fact that the term "[[quadraphonic]]" mixed Greek and Latin roots (it is a [[hybrid word]]), and that it should have properly been called "tetraphony" or "quadrasonics" (you could also call it "quadrifontal" – "four-source"). The term "ambisonics" (literally "surround
sound") does not suffer from this mongrel heritage.

In Ambisonics the term "periphony" (literally, "sound (around) the edge") is frequently used to denote full-sphere, with-height, 3-dimensional surround – note that in a periphonic system virtual sources can be localised anywhere ''within'' the sphere, not only at its surface.

Strictly speaking, we should define a difference between "with-height" and "periphony". The former implies the ability to (re)create a sensation of sounds coming from above the listener, and/or a sensation of space above the listener. "Periphony", however, strictly denotes ''full-sphere'' reproduction, which includes height ''and'' depth, providing the ability to place sounds in ''any'' direction including ''below'' the plane of the listener.

Thus a system for replaying height information might utilise a set of four speakers at ear level, say, and another four directly above them and higher up ("stacked rectangles"). This would be able to reproduce height, but not "depth". An array of "crossed rectangles", however (a horizontal rectangle at ear height and a vertical rectangle crossing it at right-angles at the centre, with two speakers at floor level and two more directly above them, ''above'' the plane of the horizontal rectangle), would permit the reproduction of depth as well as height. It is widely believed that when Michael Gerzon referred to "periphony" he meant the latter capability, as does Peter Craven, and not solely the ability to reproduce height.

The term "planar" (on a single plane, i.e. no height, or 2-dimensional) is used to refer to horizontal-only Ambisonics; the term "pantophonic" will also be found with the same meaning.

Also, in this field; "2-D" and "3-D" respectively mean planar & periphonic. It is not defined as "stereo", "5.1", etc...

=== Compass points ===
A significant difference between Ambisonics and other surround systems is that the signal is the same irrespective of the number of speakers connected to the decoder, or where they are. The decoder and speaker array do their best to ''render'' the original soundfield to the highest resolution of which the system is capable. Sound is not drawn into the speakers and you may not know where the speakers are (and it doesn't matter).

Conventional surround, however, maps one speaker to one channel. Thus each speaker (or channel) has a name based on its physical location (such as "left rear" or "right front"). In Ambisonics, it doesn't matter where the speakers are, it's the direction that's important, and the fact that the speakers are ''all'' required and working together to localise virtual sources. So we may talk about a source coming from so many degrees from centre front, and often reference is made in terms of compass points, centre front being North.

Thus while a typical surround "walk-around" or channel identification test will simply drive each speaker in turn and label the speaker from which listeners should be hearing sound, the Ambisonic equivalent will often call out compass directions, so listeners can check that the virtual source really is coming from that direction. How the points of a periphonic "fly-around" would be labelled is another matter entirely.

== See also ==

* [[Ambisonic decoding]]
* [[Ambisonic UHJ Format]]
* [[Colin McRae: DiRT]], a video game whose [[PlayStation 3]] version uses Ambisonics
* [[Colin McRae: DiRT 2]], a video game which uses Ambisonics (all versions)
* [[F1 2010 (video game)|F1 2010]], a video game which uses Ambisonics (all versions)
* [[Meridian Audio, Ltd.]]
* [[Nimbus Records]]
* [[Race Driver: GRID]], a video game whose [[PlayStation 3]] and [[Xbox 360]] versions use Ambisonics
* [[Soundfield microphone]]
* [[Surround sound]]
* [[Trifield]]

== References ==

{{Reflist|2}}

=== Source texts on Ambisonics – basic theory ===
Included with permission from the [http://members.cox.net/surround/uhjdisc/ambipubl.htm List of Ambisonic Publications], which contains an extended list of references not all included here.

{{Refbegin|2}}
* Duane H. Cooper, Takeo Shiga: ''Discrete-matrix multichannel stereo'', JAES, June 1972, Vol.20, No:5
*Michael Gerzon: ''Periphony: With-height sound reproduction'', JAES Jan/Feb. 1973, Vol.21, No:1
* Michael Gerzon: ''Surround-sound psychoacoustics, Criteria for the design of matrix and discrete surround-sound systems''. Wireless World, December 1974, pp. 483–485.
* Peter Fellgett: ''Ambisonics. Part One: General system description''. Studio Sound, August 1975, p. 20–40.
* Michael Gerzon: ''Compatible 2-channel encoding of surround sound''. NRDC reprint from Electronics Letters 11 Dec. 1975 Vol.11 Nos: 25/26.
* Michael Gerzon: ''Multidirectional sound reproduction systems'', US Patent 3,997,725. 14 Dec. 1976
* Michael Gerzon: ''The optimum choice of surround sound specification''. AES preprint No:1199, March 1977.
* Michael Gerzon: ''NRDC surround sound system''. Wireless World, April 1977, p. 36–39.
* Michael Gerzon: ''Criteria for evaluating surround sound systems''. JAES June 1977, Vol 25, No:6, p. 400–408.
* Peter Craven, Michael Gerzon: ''Coincident microphone simulation covering three dimensional space and yielding various directional outputs'', US Patent 4,042,779. 16 Aug. 1977
* Michael Gerzon: ''Sound reproductions systems with augmentation of image definition in a selected direction'', US Patent 4,081,606. 28 March 1978
* Michael Gerzon: ''Sound reproduction system with non-square loudspeaker lay-out'', US Patent 4,086,433. 25 Apr. 1978
* Michael Gerzon: ''Non-rotationally-symmetric surround-sound encoding system'', US Patent 4,095,049. 13 June 1978
* Barry Fox (writing as Adrian Hope): ''Surround sound patents, will the future of surround sound depend on patent bargaining?'' Wireless World, Jan 1979, p. 57–58.
* Michael Gerzon: ''Sound reproduction system with matrixing of power amplifier outputs'', US Patent 4,139,729. 13 Feb. 1979
* Michael Gerzon: ''Sound reproduction systems'', US Patent 4,151,369. 24 Apr. 1979
* Barry Fox (writing as Adrian Hope): ''Ambisonics – The theory and patents''. Studio Sound, Oct 1979, p. 36–44.
* Michael Gerzon: ''Practical periphony: The reproduction of full-sphere sound'', AES Preprint 1571, London 1980
* Michael Gerzon: ''Decoders for feeding irregular loudspeaker arrays'', US Patent 4,414,430. 8 Nov. 1983
* Michael Gerzon: ''Ambisonics in multichannel broadcasting and video''. JAES Vol 33, No:11, Nov. 1985 p. 859–871.
* Dermot J. Furlong: ''Comparative study of effective soundfield reconstruction''. AES preprint 2842, 18–21 Oct. 1989.
* Michael Gerzon: ''Hierarchical system of surround sound transmission for HDTV'', AES Preprint 3339, Vienna 1992
* Michael Gerzon: ''Ambisonic decoders for HDTV'', AES Preprint 3345, Vienna 1992
* W.C.Clarck, K.Alimi, B.Spendor: ''Ambisonic depending Aural recognition'', International Institute of Inuitive Audio research, IIAR 1205, pp 15–32, May 2008
{{Refend}}

== External links ==
* [http://www.ambisonic.net/ Ambisonic.net] website
* [http://members.tripod.com/martin_leese/Ambisonic/faq_latest.html Ambisonic Surround Sound FAQ]
* [http://www.ambisonia.com/ Ambisonia], a repository of Ambisonic recordings and compositions
* [http://members.cox.net/surround/uhjdisc/ambindex.htm Ambisonic Discography], a list of record releases, broadcasts and other Ambisonic content
* [http://www.ambisonia.com/wiki/ Ambisonics Wiki on Ambisonia], a knowledge base for documenting and sharing anything related to Ambisonics
* [http://members.cox.net/surround/uhjdisc/ambipubl.htm List of Ambisonic Publications], an extensive list of published references and commentaries
* [http://pcfarina.eng.unipr.it/Ambisonics.htm Ambisonics resources] at the University of Parma
* [http://www.muse.demon.co.uk/3daudio.html 3D Audio Links and Information]
* [http://www.york.ac.uk/inst/mustech/3d_audio/ Ambisonic resources] at the University of York
* [http://www.ambisonictoolkit.net/ The Ambisonic Toolkit (ATK)], software for encoding, processing and decoding Ambisonics
* [http://www.tonmeister.ca/main/textbook/intro_to_sound_recordingch11.html#x42-83400010.5.2 Why First-order Ambisonics doesn’t work]
* [http://www.ambisonia.com/wiki/index.php/Why_Ambisonics_Works Why Ambisonics Works], a short critique of the above
* [http://www.josephson.com/studio09.html Josephson Engineering], who manufacture a "native" B-Format mic, the C700S

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[[Category:Surround sound]]

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