Main Page: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
mNo edit summary
No edit summary
 
(539 intermediate revisions by more than 100 users not shown)
Line 1: Line 1:
[[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}}]]
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


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>
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
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>
* Only registered users will be able to execute this rendering mode.
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>
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.
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).
Registered users will be able to choose between the following three rendering modes:  


==Chords and modulation==
'''MathML'''
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">
:<math forcemathmode="mathml">E=mc^2</math>
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>
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


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">
'''source'''
Mathews and Pierce (1989). "The Bohlen–Pierce Scale", p.171.</ref>
:<math forcemathmode="source">E=mc^2</math> -->


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"/>
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


==Timbre and the tritave==
==Demos==
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.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


==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"
* accessibility:
!
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
| colspan="2" | '''C'''
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
| colspan="2" | '''D'''
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
| colspan="2" | '''E'''
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
| colspan="2" | '''F'''
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
| colspan="2" | '''G'''
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
| colspan="2" | '''H'''
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.
| 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}}
==Test pages ==
{{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>:
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
(5/3) (7/5)
*[[Displaystyle]]
V  IX  III
*[[MathAxisAlignment]]
      |
*[[Styling]]
    III VII I
*[[Linebreaking]]
        |
*[[Unique Ids]]
        VI I IV
*[[Help:Formula]]
          |
          IV VIII II


==Bohlen–Pierce temperament==
*[[Inputtypes|Inputtypes (private Wikis only)]]
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.
*[[Url2Image|Url2Image (private Wikis only)]]
 
==Bug reporting==
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.
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
 
==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"|&nbsp;&nbsp;0&nbsp;&nbsp;
|colspan=2 align=center bgcolor="#eeeeff"|&nbsp;133&nbsp;
|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"|<small>1018</small>
|colspan=2 align=center bgcolor="#eeeeff"|<small>1165</small>
|colspan=2 align=center bgcolor="#eeeeff"|<small>1319</small>
|colspan=2 align=center bgcolor="#eeeeff"|<small>1467</small>
|colspan=2 align=center bgcolor="#eeeeff"|<small>1600</small>
|colspan=2 align=center bgcolor="#eeeeff"|<small>1769</small>
|colspan=2 align=center bgcolor="#eeeeff"|<small>1902</small>
|}
 
'''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"|&nbsp;&nbsp;0&nbsp;&nbsp;
|colspan=2 align=center bgcolor="#eeeeff"|&nbsp;146&nbsp;
|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"|<small>1024</small>
|colspan=2 align=center bgcolor="#eeeeff"|<small>1170</small>
|colspan=2 align=center bgcolor="#eeeeff"|<small>1317</small>
|colspan=2 align=center bgcolor="#eeeeff"|<small>1463</small>
|colspan=2 align=center bgcolor="#eeeeff"|<small>1609</small>
|colspan=2 align=center bgcolor="#eeeeff"|<small>1756</small>
|colspan=2 align=center bgcolor="#eeeeff"|<small>1902</small>
|}
 
{{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]]
[[Category:Musical scales]]
[[Category:Just tunings]]
[[Category:Musical temperaments]]
 
[[de:Bohlen-Pierce-Skala]]
[[nl:Bohlen-Pierce-schaal]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

Retrieved from "https://en.formulasearchengine.com/index.php?title=Main_Page&oldid=329754"