Bohpier/Chords: Difference between revisions

<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">Below are listed the [[Dyadic chord|dyadic chords]] of 11-limit [[Sensamagic clan#Bohpier|bohpier temperament]]. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 540/539 are swetismic, by 245/243 sensamagic, by 100/99 ptolemismic, and by 1344/1331 hemimin. Chords requiring any two of 540/539, 245/243 or 100/99 are labeled octarod.

Bohpier has MOS of size 8, 9, 17, 25, 33, 41 and 49, and it may be seen that even the eight-note MOS comes equipped with some triads and tetrads. It should also be noted that the generator chain of 7-limit bohpier is the [[Bohlen-Pierce]] scale, and the same is true of 11-limit bohpier if we do not regard 11/4 as a forbidden interval because the denominator is an even number. Hence, every chord listed below has a voicing which makes it a chord of Bohlen-Pierce, showing Bohlen-Pierce contains many essentially tempered chords. The listed transversals may be converted to Bohlen-Pierce transversals by adjusting up an octave past 9/5~20/11, so that 7/6 becomes 7/3, 14/11 becomes 28/11, 11/8 becomes 11/4, 3/2 becomes 3, 18/11 becomes 36/11, 5/4 becomes 5, 7/4 becomes 7, and 9/8 becomes 9. It should also be noted that 13-limit bohpier, and hence 13-limit Bohlen-Pierce, has many more 13-limit essentially tempered chords.

In strictly traditional Bohlen-Pierce theory, only ratios with odd numbers are considered, such as produce coincident partials on instruments with only odd harmonics (e.g. an ideal clarinet). The essentially tempered chords of this 3.5.7 system are much more limited - besides the JI chords (otonal, utonal, and ambitonal), only the [[sensamagic chords]] exist in strict Bohlen-Pierce.

=Triads=  

|| Number || Chord || Transversal || Type ||

|| 1 || 0-1-4 || 1-12/11-7/5 || swetismic ||

|| 2 || 0-3-4 || 1-9/7-7/5 || swetismic ||

|| 3 || 0-3-6 || 1-9/7-5/3 || sensamagic ||

|| 4 || 0-1-7 || 1-12/11-20/11 || otonal ||

|| 5 || 0-3-7 || 1-9/7-9/5 || utonal ||

|| 6 || 0-4-7 || 1-7/5-9/5 || otonal ||

|| 7 || 0-6-7 || 1-5/3-20/11 || utonal ||

|| 8 || 0-3-10 || 1-9/7-7/6 || sensamagic ||

|| 9 || 0-4-10 || 1-7/5-7/6 || utonal ||

|| 10 || 0-6-10 || 1-5/3-7/6 || otonal ||

|| 11 || 0-7-10 || 1-9/5-7/6 || sensamagic ||

|| 12 || 0-1-11 || 1-12/11-14/11 || otonal ||

|| 13 || 0-4-11 || 1-7/5-14/11 || utonal ||

|| 14 || 0-7-11 || 1-20/11-14/11 || otonal ||

|| 15 || 0-10-11 || 1-7/6-14/11 || utonal ||

|| 16 || 0-1-12 || 1-12/11-11/8 || hemimin ||

|| 17 || 0-6-12 || 1-5/3-11/8 || ptolemismic ||

|| 18 || 0-11-12 || 1-14/11-11/8 || hemimin ||

|| 19 || 0-1-13 || 1-12/11-3/2 || utonal ||

|| 20 || 0-3-13 || 1-9/7-3/2 || utonal ||

|| 21 || 0-6-13 || 1-5/3-3/2 || otonal ||

|| 22 || 0-7-13 || 1-9/5-3/2 || utonal ||

|| 23 || 0-10-13 || 1-7/6-3/2 || otonal ||

|| 24 || 0-12-13 || 1-11/8-3/2 || otonal ||

|| 25 || 0-1-14 || 1-12/11-18/11 || otonal ||

|| 26 || 0-3-14 || 1-9/7-18/11 || utonal ||

|| 27 || 0-4-14 || 1-7/5-18/11 || swetismic ||

|| 28 || 0-7-14 || 1-9/5-18/11 || utonal ||

|| 29 || 0-10-14 || 1-7/6-18/11 || swetismic ||

|| 30 || 0-11-14 || 1-14/11-18/11 || otonal ||

|| 31 || 0-13-14 || 1-3/2-18/11 || utonal ||

|| 32 || 0-6-19 || 1-5/3-5/4 || utonal ||

|| 33 || 0-7-19 || 1-20/11-5/4 || utonal ||

|| 34 || 0-12-19 || 1-11/8-5/4 || otonal ||

|| 35 || 0-13-19 || 1-3/2-5/4 || otonal ||

|| 36 || 0-4-23 || 1-7/5-7/4 || utonal ||

|| 37 || 0-10-23 || 1-7/6-7/4 || utonal ||

|| 38 || 0-11-23 || 1-14/11-7/4 || utonal ||

|| 39 || 0-12-23 || 1-11/8-7/4 || otonal ||

|| 40 || 0-13-23 || 1-3/2-7/4 || otonal ||

|| 41 || 0-19-23 || 1-5/4-7/4 || otonal ||

|| 42 || 0-3-26 || 1-9/7-9/8 || utonal ||

|| 43 || 0-7-26 || 1-9/5-9/8 || utonal ||

|| 44 || 0-12-26 || 1-11/8-9/8 || otonal ||

|| 45 || 0-13-26 || 1-3/2-9/8 || ambitonal ||

|| 46 || 0-14-26 || 1-18/11-9/8 || utonal ||

|| 47 || 0-19-26 || 1-5/4-9/8 || otonal ||

|| 48 || 0-23-26 || 1-7/4-9/8 || otonal ||

=Tetrads=  

|| Number || Chord || Transversal || Type ||

|| 1 || 0-1-4-7 || 1-12/11-7/5-9/5 || octarod ||

|| 2 || 0-3-4-7 || 1-9/7-7/5-9/5 || swetismic ||

|| 3 || 0-3-6-7 || 1-9/7-5/3-9/5 || octarod ||

|| 4 || 0-3-4-10 || 1-9/7-7/5-7/6 || octarod ||

|| 5 || 0-3-6-10 || 1-9/7-5/3-7/6 || sensamagic ||

|| 6 || 0-3-7-10 || 1-9/7-9/5-7/6 || sensamagic ||

|| 7 || 0-4-7-10 || 1-7/5-9/5-7/6 || sensamagic ||

|| 8 || 0-6-7-10 || 1-5/3-9/5-7/6 || octarod ||

|| 9 || 0-1-4-11 || 1-12/11-7/5-14/11 || octarod ||

|| 10 || 0-1-7-11 || 1-12/11-20/11-14/11 || otonal ||

|| 11 || 0-4-7-11 || 1-7/5-9/5-14/11 || ptolemismic ||

|| 12 || 0-4-10-11 || 1-7/5-7/6-14/11 || utonal ||

|| 13 || 0-7-10-11 || 1-9/5-7/6-14/11 || octarod ||

|| 14 || 0-1-11-12 || 1-12/11-14/11-11/8 || hemimin ||

|| 15 || 0-3-6-13 || 1-9/7-5/3-3/2 || sensamagic ||

|| 16 || 0-1-7-13 || 1-12/11-9/5-3/2 || ptolemismic ||

|| 17 || 0-3-7-13 || 1-9/7-9/5-3/2 || utonal ||

|| 18 || 0-6-7-13 || 1-5/3-9/5-3/2 || ptolemismic ||

|| 19 || 0-3-10-13 || 1-9/7-7/6-3/2 || sensamagic ||

|| 20 || 0-6-10-13 || 1-5/3-7/6-3/2 || otonal ||

|| 21 || 0-7-10-13 || 1-9/5-7/6-3/2 || sensamagic ||

|| 22 || 0-1-12-13 || 1-12/11-11/8-3/2 || hemimin ||

|| 23 || 0-6-12-13 || 1-5/3-11/8-3/2 || ptolemismic ||

|| 24 || 0-1-4-14 || 1-12/11-7/5-18/11 || swetismic ||

|| 25 || 0-3-4-14 || 1-9/7-7/5-18/11 || swetismic ||

|| 26 || 0-1-7-14 || 1-12/11-20/11-18/11 || otonal ||

|| 27 || 0-3-7-14 || 1-9/7-9/5-18/11 || utonal ||

|| 28 || 0-4-7-14 || 1-7/5-9/5-18/11 || octarod ||

|| 29 || 0-3-10-14 || 1-9/7-7/6-18/11 || octarod ||

|| 30 || 0-4-10-14 || 1-7/5-7/6-18/11 || swetismic ||

|| 31 || 0-7-10-14 || 1-9/5-7/6-18/11 || octarod ||

|| 32 || 0-1-11-14 || 1-12/11-14/11-18/11 || otonal ||

|| 33 || 0-4-11-14 || 1-7/5-14/11-18/11 || octarod ||

|| 34 || 0-7-11-14 || 1-20/11-14/11-18/11 || otonal ||

|| 35 || 0-10-11-14 || 1-7/6-14/11-18/11 || swetismic ||

|| 36 || 0-1-13-14 || 1-12/11-3/2-18/11 || ambitonal ||

|| 37 || 0-3-13-14 || 1-9/7-3/2-18/11 || utonal ||

|| 38 || 0-7-13-14 || 1-9/5-3/2-18/11 || utonal ||

|| 39 || 0-10-13-14 || 1-7/6-3/2-18/11 || swetismic ||

|| 40 || 0-6-7-19 || 1-5/3-20/11-5/4 || utonal ||

|| 41 || 0-6-12-19 || 1-5/3-11/8-5/4 || ptolemismic ||

|| 42 || 0-6-13-19 || 1-5/3-3/2-5/4 || ambitonal ||

|| 43 || 0-7-13-19 || 1-9/5-3/2-5/4 || ptolemismic ||

|| 44 || 0-12-13-19 || 1-11/8-3/2-5/4 || otonal ||

|| 45 || 0-4-10-23 || 1-7/5-7/6-7/4 || utonal ||

|| 46 || 0-4-11-23 || 1-7/5-14/11-7/4 || utonal ||

|| 47 || 0-10-11-23 || 1-7/6-14/11-7/4 || utonal ||

|| 48 || 0-11-12-23 || 1-14/11-11/8-7/4 || hemimin ||

|| 49 || 0-10-13-23 || 1-7/6-3/2-7/4 || ambitonal ||

|| 50 || 0-12-13-23 || 1-11/8-3/2-7/4 || otonal ||

|| 51 || 0-12-19-23 || 1-11/8-5/4-7/4 || otonal ||

|| 52 || 0-13-19-23 || 1-3/2-5/4-7/4 || otonal ||

|| 53 || 0-3-7-26 || 1-9/7-9/5-9/8 || utonal ||

|| 54 || 0-3-13-26 || 1-9/7-3/2-9/8 || utonal ||

|| 55 || 0-7-13-26 || 1-9/5-3/2-9/8 || utonal ||

|| 56 || 0-12-13-26 || 1-11/8-3/2-9/8 || otonal ||

|| 57 || 0-3-14-26 || 1-9/7-18/11-9/8 || utonal ||

|| 58 || 0-7-14-26 || 1-9/5-18/11-9/8 || utonal ||

|| 59 || 0-13-14-26 || 1-3/2-18/11-9/8 || utonal ||

|| 60 || 0-7-19-26 || 1-9/5-5/4-9/8 || ptolemismic ||

|| 61 || 0-12-19-26 || 1-11/8-5/4-9/8 || otonal ||

|| 62 || 0-13-19-26 || 1-3/2-5/4-9/8 || otonal ||

|| 63 || 0-12-23-26 || 1-11/8-7/4-9/8 || otonal ||

|| 64 || 0-13-23-26 || 1-3/2-7/4-9/8 || otonal ||

|| 65 || 0-19-23-26 || 1-5/4-7/4-9/8 || otonal ||

=Pentads=  

=Hexads=

|| Number || Chord || Transversal || Type ||

|| 1 || 0-3-6-7-10-13 || 1-9/7-5/3-9/5-7/6-3/2 || octarod ||

|| 2 || 0-3-4-7-10-14 || 1-9/7-7/5-9/5-7/6-18/11 || octarod ||

|| 3 || 0-1-4-7-11-14 || 1-12/11-7/5-9/5-14/11-18/11 || octarod ||

|| 4 || 0-4-7-10-11-14 || 1-7/5-9/5-7/6-14/11-18/11 || octarod ||

|| 5 || 0-3-7-10-13-14 || 1-9/7-9/5-7/6-3/2-18/11 || octarod ||

|| 6 || 0-3-7-13-14-26 || 1-9/7-9/5-3/2-18/11-9/8 || utonal ||

|| 7 || 0-12-13-19-23-26 || 1-11/8-3/2-5/4-7/4-9/8 || otonal ||</pre></div>

<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;Chords of bohpier&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Below are listed the &lt;a class="wiki_link" href="/Dyadic%20chord"&gt;dyadic chords&lt;/a&gt; of 11-limit &lt;a class="wiki_link" href="/Sensamagic%20clan#Bohpier"&gt;bohpier temperament&lt;/a&gt;. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 540/539 are swetismic, by 245/243 sensamagic, by 100/99 ptolemismic, and by 1344/1331 hemimin. Chords requiring any two of 540/539, 245/243 or 100/99 are labeled octarod.&lt;br /&gt;

Bohpier has MOS of size 8, 9, 17, 25, 33, 41 and 49, and it may be seen that even the eight-note MOS comes equipped with some triads and tetrads. It should also be noted that the generator chain of 7-limit bohpier is the &lt;a class="wiki_link" href="/Bohlen-Pierce"&gt;Bohlen-Pierce&lt;/a&gt; scale, and the same is true of 11-limit bohpier if we do not regard 11/4 as a forbidden interval because the denominator is an even number. Hence, every chord listed below has a voicing which makes it a chord of Bohlen-Pierce, showing Bohlen-Pierce contains many essentially tempered chords. The listed transversals may be converted to Bohlen-Pierce transversals by adjusting up an octave past 9/5~20/11, so that 7/6 becomes 7/3, 14/11 becomes 28/11, 11/8 becomes 11/4, 3/2 becomes 3, 18/11 becomes 36/11, 5/4 becomes 5, 7/4 becomes 7, and 9/8 becomes 9. It should also be noted that 13-limit bohpier, and hence 13-limit Bohlen-Pierce, has many more 13-limit essentially tempered chords.&lt;br /&gt;

&lt;br /&gt;

In strictly traditional Bohlen-Pierce theory, only ratios with odd numbers are considered, such as produce coincident partials on instruments with only odd harmonics (e.g. an ideal clarinet). The essentially tempered chords of this 3.5.7 system are much more limited - besides the JI chords (otonal, utonal, and ambitonal), only the &lt;a class="wiki_link" href="/sensamagic%20chords"&gt;sensamagic chords&lt;/a&gt; exist in strict Bohlen-Pierce.&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Triads"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Triads&lt;/h1&gt;

&lt;table class="wiki_table"&gt;

        &lt;td&gt;Number&lt;br /&gt;

        &lt;td&gt;Chord&lt;br /&gt;

        &lt;td&gt;Transversal&lt;br /&gt;

        &lt;td&gt;Type&lt;br /&gt;

        &lt;td&gt;0-1-4&lt;br /&gt;

        &lt;td&gt;1-12/11-7/5&lt;br /&gt;

        &lt;td&gt;0-3-4&lt;br /&gt;

        &lt;td&gt;1-9/7-7/5&lt;br /&gt;

        &lt;td&gt;0-3-6&lt;br /&gt;

        &lt;td&gt;1-9/7-5/3&lt;br /&gt;

        &lt;td&gt;1-9/7-9/5&lt;br /&gt;

        &lt;td&gt;1-7/5-9/5&lt;br /&gt;

        &lt;td&gt;7&lt;br /&gt;

        &lt;td&gt;0-6-7&lt;br /&gt;

        &lt;td&gt;1-5/3-20/11&lt;br /&gt;

        &lt;td&gt;0-3-10&lt;br /&gt;

        &lt;td&gt;0-4-10&lt;br /&gt;

        &lt;td&gt;1-7/5-7/6&lt;br /&gt;

        &lt;td&gt;10&lt;br /&gt;

        &lt;td&gt;0-6-10&lt;br /&gt;

        &lt;td&gt;1-5/3-7/6&lt;br /&gt;

        &lt;td&gt;0-7-10&lt;br /&gt;

        &lt;td&gt;1-9/5-7/6&lt;br /&gt;

        &lt;td&gt;0-1-11&lt;br /&gt;

        &lt;td&gt;1-12/11-14/11&lt;br /&gt;

        &lt;td&gt;0-4-11&lt;br /&gt;

        &lt;td&gt;0-7-11&lt;br /&gt;

        &lt;td&gt;1-20/11-14/11&lt;br /&gt;

        &lt;td&gt;1-7/6-14/11&lt;br /&gt;

        &lt;td&gt;1-12/11-11/8&lt;br /&gt;

        &lt;td&gt;1-14/11-11/8&lt;br /&gt;

        &lt;td&gt;1-12/11-3/2&lt;br /&gt;

        &lt;td&gt;1-9/7-3/2&lt;br /&gt;

&lt;/td&gt;

        &lt;td&gt;1-5/3-3/2&lt;br /&gt;

        &lt;td&gt;1-9/5-3/2&lt;br /&gt;

        &lt;td&gt;0-1-14&lt;br /&gt;

        &lt;td&gt;1-12/11-18/11&lt;br /&gt;

        &lt;td&gt;0-3-14&lt;br /&gt;

        &lt;td&gt;1-9/7-18/11&lt;br /&gt;

        &lt;td&gt;0-4-14&lt;br /&gt;

        &lt;td&gt;1-7/5-18/11&lt;br /&gt;

        &lt;td&gt;1-9/5-18/11&lt;br /&gt;

        &lt;td&gt;1-7/6-18/11&lt;br /&gt;

        &lt;td&gt;1-14/11-18/11&lt;br /&gt;

        &lt;td&gt;0-13-14&lt;br /&gt;

        &lt;td&gt;0-6-19&lt;br /&gt;

        &lt;td&gt;1-5/3-5/4&lt;br /&gt;

        &lt;td&gt;0-7-19&lt;br /&gt;

        &lt;td&gt;0-13-19&lt;br /&gt;

        &lt;td&gt;1-3/2-5/4&lt;br /&gt;

        &lt;td&gt;1-7/5-7/4&lt;br /&gt;