Bohlen–Pierce scale: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
|en = Bohlen-Pierce
: This revision was by author [[User:vaisvil|vaisvil]] and made on <tt>2012-01-03 16:51:35 UTC</tt>.<br>
|de = Bohlen-Pierce
: The original revision id was <tt>289364899</tt>.<br>
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: The revision comment was: <tt></tt><br>
{{Wikipedia}}
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
[[File:vaisvil-BP-guitar-052-crop.jpg|thumb|x00px|Chris Vaisvil's BP electric guitar. [http://chrisvaisvil.com/the-bohlen-pierce-epiphone-roadie-guitar/ Music from this guitar.]]]
<h4>Original Wikitext content:</h4>
[[File:Sword_BP_guitars.jpg|thumb|300px]]
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=The Bohlen-Pierce scale=


The '''Bohlen–Pierce scale''' ('''BP''') is a 13-tone [[macrotonal]] [[nonoctave]] [[scale]] designed to emphasize odd-number intervals and chords, such as the 3:5:7:9 tetrad. It was first described as a 7-limit [[just intonation]] scale and as an [[equal-step tuning|equal temperament]], [[13edt|13 equal divisions of the tritave]]. The [[tritave]] (3/1) usually replaces the [[octave]] in the role of the [[equave]], such that intervals a tritave apart are considered [[equivalent]].


The **Bohlen-Pierce** (**BP**) scale is a [[nonoctave]] scale, a 13-part equal division of the perfect-twelfth ([[3_1|3/1]]) or [[Tritave]] (**13edt**). Each step is about 146 ¢, making it a [[macrotonal]] scale. It is closely related to the rank two temperament [[Sensamagic clan#Bohpier|bohpier]], and in fact chords of Bohlen-Pierce may be found listed on the page [[chords of bohpier]].
It is closely related to the rank two temperament [[Sensamagic clan #Bohpier|bohpier]]. It is normally thought of (if not in these terms, then in fact) as a temperament defined on the 3.5.7 [[Just intonation subgroup|subgroup]]. However, it can be extended to the 3.5.7.11/4 subgroup, especially when considering 13edt instead of the JI version. This extension is controversial because of the presence of 2 in the denominator of 11/4, but the interval is present in the sense that 3^(12\13) provides an approximation to it. Chords of Bohlen–Pierce, from this extended perspective, may be found listed on the page [[chords of bohpier]].  


It was discovered independently by [[Heinz Bohlen]], [[John Pierce]], [[Kees van Prooijen]], and perhaps others, usually noticed for its good approximation of odd-number just ratios 3:5, 5:7, 3:7, etc.
Bohlen–Pierce was discovered independently by [[Heinz Bohlen]], [[John Pierce]], [[Kees van Prooijen]], and perhaps others, usually noticed for its good approximation of odd-number just ratios 3:5, 5:7, 3:7, etc.; but not necessarily 4:11, 5:6, 6:7, etc.


== Theory ==
* [[Relationship between Bohlen–Pierce and octave-ful temperaments]]


=== Lambda scale ===
{{Main|4L 5s (3/1-equivalent)}}


[[image:http://www.ronsword.com/images/Sword_BP_guitars.jpg width="543" height="422"]]
== Intervals ==
{{Main|Intervals of BP}}


[[image:http://ronsword.com/bpguitar.JPG width="171" height="224"]]
== Variations ==
[[Ron Sword]] and his 9-string 13-tone BP Touchstick/Guitar crossover instrument (aka: "//the Bohlen-Box//")
=== Triple Bohlen–Pierce ===
Proposed by [[Paul Erlich]] is [[39edt]], also known as the ''Triple Bohlen–Pierce scale''. It adds to 13edt accurate approximations to the 11th and 13th harmonics ([[11/9]] and [[13/9]]) and can be used in a variety of ways, for both just intonation chords and harmonies, as standard Bohlen–Pierce scale interlocking three times with calm-sounding quarter-tones, and for various JI modulations.


**Triple Bohlen-Pierce**
== Regular temperament properties ==
{{Main| 13edt#Regular temperament properties }}


Proposed by [[Paul Erlich]], is the Triple Bohlen-Pierce Scale, or 39th root of 3. It approximates additional odd harmonics and can be used in a variety of ways, for both just intonation chords and harmonies, as standard Bohlen-Pierce scale interlocking three times with calm sounding quarter-tones, and for various JI modulations.
== Instruments ==
* [http://www.huygens-fokker.org/bpsite/instruments.html#anchor39641 Bohlen Pierce guitar]
* [[Clarinet#Bohlen-Pierce clarinets|Clarinets]]
* Metallophone
* Electronic Organ
* Stredici
* [http://bp.b0b.com/2016/01/tuning-a-kalimba-to-bp/ Kalimba] (Mbira)
* [http://bp.b0b.com/2012/11/sierra-pedal-steel/ Pedal Steel Guitar]


[[image:http://www.ronsword.com/images/3BP3.JPG width="672" height="340"]](triple bohlen-pierce &lt;span class="UIStory_Message"&gt;39√3 Classical Guitar by Ron Sword&lt;/span&gt;)
== Music ==
**Theory**
{{Main| Bohlen–Pierce scale/Music }}
[[http://www.huygens-fokker.org/bpsite/|Bohlen Pierce website]]
{{Catrel|Bohlen-Pierce tracks}}
[[http://en.wikipedia.org/wiki/Bohlen-Pierce_scale|Wikipedia Bohlen-Pierce scale]]
 
[[http://ziaspace.com/_microtonality/BP/|Bohlen-Pierce Scale Research]] by Elaine Walker
== See also ==
Sword, Ronald. "Bohlen Pierce Scales for Guitar" IAAA Press, UK-USA. First Ed: May 2009.
* [[Catalog of 3.5.7 subgroup rank two temperaments]]
[[Intervals of BP]]
* [[Lumatone mapping for Bohlen–Pierce]]
==Physical instruments tuned to the BP scale==  
 
[[@http://www.swordguitars.com/Sword_Bohlen_Pierce_acoustic.JPG|Bohlen Pierce guitar]]
== Further reading ==
[[microtonal clarinet#BPClarinets|Clarinets]]
* [[Sword, Ron]]. [https://ronsword.bigcartel.com/product/creative-applications-for-polyscales-and-scale-sequences ''Bohlen-Pierce Scales for Guitar: Includes Paul Erlich's Triple BP Scales. Notation methods, Chord-Scales, Melody Chords, Polyscales, and new Scales for the Bohlen-Pierce tuning'']. 2009. ([http://www.metatonalmusic.com/books.html Metatonal Music link])
Metallophone
 
Electronic Organ
== External links ==
Stredici
* [http://www.huygens-fokker.org/bpsite/ The Bohlen-Pierce Site]
==Compositions==  
* [http://ziaspace.com/_academic/BP/ Bohlen-Pierce Scale Research] by [[Elaine Walker]]
Links to available music written in BP at [[http://www.huygens-fokker.org/bpsite/references2.html#anchor81501|above website]].
 
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;[[http://www.seraph.it/XenoTunes4_files/Bohl-en%20Roll.mp3|Bohl-en Roll]]&lt;/span&gt; by [[Carlo Serafini]]
[[Category:Bohlen–Pierce| ]] <!-- main article -->
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;[[@http://www.jeanpierrepoulin.com/mp3/BPguitares.mp3|Bohlen-Pierce electric guitar improvisation]]&lt;/span&gt; by [[@http://www.jeanpierrepoulin.com/|Jean-Pierre Poulin]]
[[Category:13-tone scales]]
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;[[@http://www.ronsword.com/sounds/BP_stretched_chroma.mp3|Bohlen-Pierce "Stretched Chroma" Acoustic Improvisation]]&lt;/span&gt; by [[Ron Sword]]
[[Category:Guitar]]
[[@http://www.kunstmusik.com/2010/03/08/reminiscences/|Reminiscences]] [[Steven Yi]]
[[Category:Macrotonal]]
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;[[http://www.jeanpierrepoulin.com/mp3/Roll%27n%27Peace.mp3|Roll'n'Peace]]&lt;/span&gt; by Jean-Pierre Poulin[[media type="custom" key="11371948"]]
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/01%20-%201.%20Bohlen-Pierce.mp3|Comets Over Flatland 1]]&lt;/span&gt; by [[Randy Winchester]]
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/02%20-%202.%20Bohlen-Pierce.mp3|Comets Over Flatland 2]]&lt;/span&gt; by [[Randy Winchester]]
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/03%20-%203.%20Bohlen-Pierce.mp3|Comets Over Flatland 3]]&lt;/span&gt; by [[Randy Winchester]]
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/04%20-%204.%20Bohlen-Pierce.mp3|Comets Over Flatland 4]]&lt;/span&gt; by [[Randy Winchester]]
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;[[http://micro.soonlabel.com/bohlen-pierce/daily20110915_BP_strings-Bohlen-Pierce-Island.mp3|Bohlen-Pierce Island audio]]&lt;/span&gt; by [[Chris Vaisvil]]
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;[[http://micro.soonlabel.com/bohlen-pierce/mesonic_atom_13th_root_of_3_BP_apx_X1.mp3|Mesonic Atom]]&lt;/span&gt; by [[Chris Vaisvil]]
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;[[http://micro.soonlabel.com/MOTU/Vaisvil-MOTU-8a-bending-the-rules-in-bohlen-pierce.mp3|Bending the Rules]]&lt;/span&gt; by [[Chris Vaisvil]]</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Bohlen-Pierce&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:1:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="The Bohlen-Pierce scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:1 --&gt;The Bohlen-Pierce scale&lt;/h1&gt;
&lt;br /&gt;
&lt;br /&gt;
The &lt;strong&gt;Bohlen-Pierce&lt;/strong&gt; (&lt;strong&gt;BP&lt;/strong&gt;) scale is a &lt;a class="wiki_link" href="/nonoctave"&gt;nonoctave&lt;/a&gt; scale, a 13-part equal division of the perfect-twelfth (&lt;a class="wiki_link" href="/3_1"&gt;3/1&lt;/a&gt;) or &lt;a class="wiki_link" href="/Tritave"&gt;Tritave&lt;/a&gt; (&lt;strong&gt;13edt&lt;/strong&gt;). Each step is about 146 ¢, making it a &lt;a class="wiki_link" href="/macrotonal"&gt;macrotonal&lt;/a&gt; scale. It is closely related to the rank two temperament &lt;a class="wiki_link" href="/Sensamagic%20clan#Bohpier"&gt;bohpier&lt;/a&gt;, and in fact chords of Bohlen-Pierce may be found listed on the page &lt;a class="wiki_link" href="/chords%20of%20bohpier"&gt;chords of bohpier&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
It was discovered independently by &lt;a class="wiki_link" href="/Heinz%20Bohlen"&gt;Heinz Bohlen&lt;/a&gt;, &lt;a class="wiki_link" href="/John%20Pierce"&gt;John Pierce&lt;/a&gt;, &lt;a class="wiki_link" href="/Kees%20van%20Prooijen"&gt;Kees van Prooijen&lt;/a&gt;, and perhaps others, usually noticed for its good approximation of odd-number just ratios 3:5, 5:7, 3:7, etc.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextRemoteImageRule:7:&amp;lt;img src=&amp;quot;http://www.ronsword.com/images/Sword_BP_guitars.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 422px; width: 543px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.ronsword.com/images/Sword_BP_guitars.jpg" alt="external image Sword_BP_guitars.jpg" title="external image Sword_BP_guitars.jpg" style="height: 422px; width: 543px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:7 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextRemoteImageRule:8:&amp;lt;img src=&amp;quot;http://ronsword.com/bpguitar.JPG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 224px; width: 171px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://ronsword.com/bpguitar.JPG" alt="external image bpguitar.JPG" title="external image bpguitar.JPG" style="height: 224px; width: 171px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:8 --&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/Ron%20Sword"&gt;Ron Sword&lt;/a&gt; and his 9-string 13-tone BP Touchstick/Guitar crossover instrument (aka: &amp;quot;&lt;em&gt;the Bohlen-Box&lt;/em&gt;&amp;quot;)&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Triple Bohlen-Pierce&lt;/strong&gt;&lt;br /&gt;
&lt;br /&gt;
Proposed by &lt;a class="wiki_link" href="/Paul%20Erlich"&gt;Paul Erlich&lt;/a&gt;, is the Triple Bohlen-Pierce Scale, or 39th root of 3. It approximates additional odd harmonics and can be used in a variety of ways, for both just intonation chords and harmonies, as standard Bohlen-Pierce scale interlocking three times with calm sounding quarter-tones, and for various JI modulations.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextRemoteImageRule:9:&amp;lt;img src=&amp;quot;http://www.ronsword.com/images/3BP3.JPG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 340px; width: 672px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.ronsword.com/images/3BP3.JPG" alt="external image 3BP3.JPG" title="external image 3BP3.JPG" style="height: 340px; width: 672px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:9 --&gt;(triple bohlen-pierce &lt;span class="UIStory_Message"&gt;39√3 Classical Guitar by Ron Sword&lt;/span&gt;)&lt;br /&gt;
&lt;strong&gt;Theory&lt;/strong&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.huygens-fokker.org/bpsite/" rel="nofollow"&gt;Bohlen Pierce website&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Bohlen-Pierce_scale" rel="nofollow"&gt;Wikipedia Bohlen-Pierce scale&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://ziaspace.com/_microtonality/BP/" rel="nofollow"&gt;Bohlen-Pierce Scale Research&lt;/a&gt; by Elaine Walker&lt;br /&gt;
Sword, Ronald. &amp;quot;Bohlen Pierce Scales for Guitar&amp;quot; IAAA Press, UK-USA. First Ed: May 2009.&lt;br /&gt;
&lt;a class="wiki_link" href="/Intervals%20of%20BP"&gt;Intervals of BP&lt;/a&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="The Bohlen-Pierce scale-Physical instruments tuned to the BP scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;Physical instruments tuned to the BP scale&lt;/h2&gt;
&lt;a class="wiki_link_ext" href="http://www.swordguitars.com/Sword_Bohlen_Pierce_acoustic.JPG" rel="nofollow" target="_blank"&gt;Bohlen Pierce guitar&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/microtonal%20clarinet#BPClarinets"&gt;Clarinets&lt;/a&gt;&lt;br /&gt;
Metallophone&lt;br /&gt;
Electronic Organ&lt;br /&gt;
Stredici&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="The Bohlen-Pierce scale-Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;Compositions&lt;/h2&gt;
Links to available music written in BP at &lt;a class="wiki_link_ext" href="http://www.huygens-fokker.org/bpsite/references2.html#anchor81501" rel="nofollow"&gt;above website&lt;/a&gt;.&lt;br /&gt;
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes4_files/Bohl-en%20Roll.mp3" rel="nofollow"&gt;Bohl-en Roll&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt;&lt;br /&gt;
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;a class="wiki_link_ext" href="http://www.jeanpierrepoulin.com/mp3/BPguitares.mp3" rel="nofollow" target="_blank"&gt;Bohlen-Pierce electric guitar improvisation&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link_ext" href="http://www.jeanpierrepoulin.com/" rel="nofollow" target="_blank"&gt;Jean-Pierre Poulin&lt;/a&gt;&lt;br /&gt;
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;a class="wiki_link_ext" href="http://www.ronsword.com/sounds/BP_stretched_chroma.mp3" rel="nofollow" target="_blank"&gt;Bohlen-Pierce &amp;quot;Stretched Chroma&amp;quot; Acoustic Improvisation&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Ron%20Sword"&gt;Ron Sword&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.kunstmusik.com/2010/03/08/reminiscences/" rel="nofollow" target="_blank"&gt;Reminiscences&lt;/a&gt; &lt;a class="wiki_link" href="/Steven%20Yi"&gt;Steven Yi&lt;/a&gt;&lt;br /&gt;
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;a class="wiki_link_ext" href="http://www.jeanpierrepoulin.com/mp3/Roll%27n%27Peace.mp3" rel="nofollow"&gt;Roll'n'Peace&lt;/a&gt;&lt;/span&gt; by Jean-Pierre Poulin&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/11371948?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;11371948&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;&lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&gt;
&lt;/script&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt;&lt;br /&gt;
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/01%20-%201.%20Bohlen-Pierce.mp3" rel="nofollow"&gt;Comets Over Flatland 1&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Randy%20Winchester"&gt;Randy Winchester&lt;/a&gt;&lt;br /&gt;
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/02%20-%202.%20Bohlen-Pierce.mp3" rel="nofollow"&gt;Comets Over Flatland 2&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Randy%20Winchester"&gt;Randy Winchester&lt;/a&gt;&lt;br /&gt;
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/03%20-%203.%20Bohlen-Pierce.mp3" rel="nofollow"&gt;Comets Over Flatland 3&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Randy%20Winchester"&gt;Randy Winchester&lt;/a&gt;&lt;br /&gt;
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/04%20-%204.%20Bohlen-Pierce.mp3" rel="nofollow"&gt;Comets Over Flatland 4&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Randy%20Winchester"&gt;Randy Winchester&lt;/a&gt;&lt;br /&gt;
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/bohlen-pierce/daily20110915_BP_strings-Bohlen-Pierce-Island.mp3" rel="nofollow"&gt;Bohlen-Pierce Island audio&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt;&lt;br /&gt;
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/bohlen-pierce/mesonic_atom_13th_root_of_3_BP_apx_X1.mp3" rel="nofollow"&gt;Mesonic Atom&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt;&lt;br /&gt;
&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/MOTU/Vaisvil-MOTU-8a-bending-the-rules-in-bohlen-pierce.mp3" rel="nofollow"&gt;Bending the Rules&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 15:43, 31 July 2025

English Wikipedia has an article on:
Chris Vaisvil's BP electric guitar. Music from this guitar.

The Bohlen–Pierce scale (BP) is a 13-tone macrotonal nonoctave scale designed to emphasize odd-number intervals and chords, such as the 3:5:7:9 tetrad. It was first described as a 7-limit just intonation scale and as an equal temperament, 13 equal divisions of the tritave. The tritave (3/1) usually replaces the octave in the role of the equave, such that intervals a tritave apart are considered equivalent.

It is closely related to the rank two temperament bohpier. It is normally thought of (if not in these terms, then in fact) as a temperament defined on the 3.5.7 subgroup. However, it can be extended to the 3.5.7.11/4 subgroup, especially when considering 13edt instead of the JI version. This extension is controversial because of the presence of 2 in the denominator of 11/4, but the interval is present in the sense that 3^(12\13) provides an approximation to it. Chords of Bohlen–Pierce, from this extended perspective, may be found listed on the page chords of bohpier.

Bohlen–Pierce was discovered independently by Heinz Bohlen, John Pierce, Kees van Prooijen, and perhaps others, usually noticed for its good approximation of odd-number just ratios 3:5, 5:7, 3:7, etc.; but not necessarily 4:11, 5:6, 6:7, etc.

Theory

Lambda scale

Main article: 4L 5s (3/1-equivalent)

Intervals

Main article: Intervals of BP

Variations

Triple Bohlen–Pierce

Proposed by Paul Erlich is 39edt, also known as the Triple Bohlen–Pierce scale. It adds to 13edt accurate approximations to the 11th and 13th harmonics (11/9 and 13/9) and can be used in a variety of ways, for both just intonation chords and harmonies, as standard Bohlen–Pierce scale interlocking three times with calm-sounding quarter-tones, and for various JI modulations.

Regular temperament properties

Main article: 13edt § Regular temperament properties

Instruments

Music

Main article: Bohlen–Pierce scale/Music
See also: Category:Bohlen-Pierce tracks

See also

Further reading

External links

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