15edo: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-11-29 01:56:47 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-11-29 02:47:08 UTC</tt>.

: The original revision id was <tt>~~600700440~~</tt>.

: The original revision id was <tt>600702192</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Line 75:

*based on treating 15-EDO as an 11-limit temperament; other approaches are possible

In [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|ups and downs notation]], which is fifth-generated, every 15edo note has at least three names. 15edo can be notated ~~without ups and downs by~~ using the [[Ups and Downs Notation#Natural%20Generators|natural generator]], which is not the 9\15 5th but the 2\15 2nd, also known as porcupine notation. The 15edo porcupine genchain is:

In [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|ups and downs notation]], which is fifth-generated, every 15edo note has at least three names. 15edo can also be notated using the [[Ups and Downs Notation#Natural%20Generators|natural generator]], which is not the 9\15 5th but the 2\15 2nd, also known as porcupine notation. The 15edo porcupine genchain is:

...A6 - A7 - A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8 - d2 - d3...

||= EDOstep ||= cents ||= ups and downs

||= EDOstep ||= cents ||= ups and downs relative notation

relative notation ||= ups and downs

(partial list, e.g. M2 is also A1 and d4) ||= ups and downs

absolute notation ||= porcupine

relative notation ||= porcupine

Line 85:

||= 0 ||= 0¢ ||= unison, min 2nd ||= C# / D / Eb ||= unison ||= D ||

||= 1 ||= 80¢ ||= up-unison, upminor 2nd ||= C#^ / D^ / Eb^ ||= aug unison, dim 2nd ||= D# / Eb ||

||= 2 ||= 160¢ ||= downaug 1, downmajor 2nd~~, downminor 3rd~~ ||= D#v / Ev / Fv / Gbv ||= perf 2nd ||= E ||

||= 2 ||= 160¢ ||= downaug unison, downmajor 2nd ||= D#v / Ev / Fv / Gbv ||= perf 2nd ||= E ||

||= 3 ||= 240¢ ||= ~~aug 1,~~ maj 2nd, min 3rd~~, dim 4th~~ ||= D# / E / F / Gb ||= aug 2nd, dim 3rd ||= E# / Fb ||

||= 3 ||= 240¢ ||= maj 2nd, min 3rd ||= D# / E / F / Gb ||= aug 2nd, dim 3rd ||= E# / Fb ||

||= 4 ||= 320¢ ||= ~~upaug 1, upmaj 2nd,~~ upmin 3rd~~, updim 4th~~ ||= D#^ / E^ / F^ / Gb^ ||= min 3rd ||= F ||

||= 4 ||= 320¢ ||= upmin 3rd ||= D#^ / E^ / F^ / Gb^ ||= min 3rd ||= F ||

||= 5 ||= 400¢ ||= downmaj 3rd~~, down 4th, downdim 5th~~ ||= F#v / Gv / Abv ||= maj 3rd ||= F# / Gb ||

||= 5 ||= 400¢ ||= downmaj 3rd ||= F#v / Gv / Abv ||= maj 3rd ||= F# / Gb ||

||= 6 ||= 480¢ ||= maj 3rd, perf 4th, dim 5th ||= F# / G / Ab ||= min 4th ||= G ||

||= 7 ||= 560¢ ||= ~~upmaj 3rd,~~ up 4th, updim 5th ||= F#^ / G^ / Ab^ ||= maj 4th ||= G# / Abb ||

||= 7 ||= 560¢ ||= up 4th, updim 5th ||= F#^ / G^ / Ab^ ||= maj 4th ||= G# / Abb ||

||= 8 ||= 640¢ ||= downaug 4th, down 5th~~, downmin 6th~~ ||= G#v / Av / Bbv ||= min 5th ||= Gx / Ab ||

||= 8 ||= 640¢ ||= downaug 4th, down 5th ||= G#v / Av / Bbv ||= min 5th ||= Gx / Ab ||

||= 9 ||= 720¢ ||= aug 4th, perf 5th, min 6th ||= G# / A / Bb ||= maj 5th ||= A ||

||= 10 ||= 800¢ ||= ~~upaug 4th,~~ up 5th, upmin 6th ||= G#^ / A^ / Bb^ ||= min 6th ||= A# / Bb ||

||= 10 ||= 800¢ ||= up 5th, upmin 6th ||= G#^ / A^ / Bb^ ||= min 6th ||= A# / Bb ||

||= 11 ||= 880¢ ||= downaug 5th, downmaj 6th~~, downmin 7th~~ ||= A#v / Bv / Cv / Dbv ||= maj 6th ||= B ||

||= 11 ||= 880¢ ||= downaug 5th, downmaj 6th ||= A#v / Bv / Cv / Dbv ||= maj 6th ||= B ||

||= 12 ||= 960¢ ||= ~~aug 5th,~~ maj 6th, min 7th~~, dim 8ve~~ ||= A# / B / C / Db ||= aug 6th, dim 7th ||= B# / Cb ||

||= 12 ||= 960¢ ||= maj 6th, min 7th ||= A# / B / C / Db ||= aug 6th, dim 7th ||= B# / Cb ||

||= 13 ||= 1040¢ ||= ~~upmaj 6th,~~ upmin 7th~~, updim 8ve~~ ||= A#^ / B^ / C^ / Db^ ||= perf 7th ||= C ||

||= 13 ||= 1040¢ ||= upmin 7th ||= A#^ / B^ / C^ / Db^ ||= perf 7th ||= C ||

||= 14 ||= 1120¢ ||= downmaj 7th, down octave~~, downmin 9th~~ ||= C#v / Dv / Ebv ||= aug 7th, dim 8ve ||= C# / Db ||

||= 14 ||= 1120¢ ||= downmaj 7th, down octave ||= C#v / Dv / Ebv ||= aug 7th, dim 8ve ||= C# / Db ||

||= 15 ||= 1200¢ ||= maj 7th, octave, min 9th ||= C# / D / Eb ||= 8ve ||= D ||

15edo chord names using ups and downs:

0-3-9 = D F A = Dm = "D minor" (or possibly D E A = Dsus2)

0-4-9 = D F^ A = D.^m = "D upminor"

0-5-9 = D F#v A = D.v = "D dot down" or "D downmajor"

0-6-9 = D F# A = D = "D" or "D major" (or possibly D G A = Dsus4)

0-3-9-12 = D F A C = Dm7 = "D minor seven", or D F A B = Dm6 = "D minor six"

0-4-9-12 = D F^ A C = Dm7(^3) = "D minor seven up-three", or D F^ A B = Dm6(^3) = "D minor six up-three"

0-5-9-12 = D F#v A C = D7(v3) = "D seven down-three", or D F#v A B = D6(v3) = "D six down-three"

0-6-9-12 = D F# A C = D7 = "D seven", or D F# A B = D6 = "D six"

0-5-9-14 = D F#v A C#v = D.vM7 = "D downmajor seven"

0-4-9-13 = D F^ A C^ = D.^m7 = "D dot up minor-seven", or D F^ A B^ = D.^m6 = "D dot up minor-six"

For a more complete list, see [[Ups and Downs Notation#Chord%20names%20in%20other%20EDOs|Ups and Downs Notation - Chord names in other EDOs]]. Because many intervals have several names, many chords do too.

15-EDO offers some minor improvements over 12-TET in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to the 7th and 11th harmonics, but its approximation to the 3rd harmonic is rather off. However, the particular way in which this approximation is off is as much a feature as it is a bug, for it allows the construction of a 5L5s MOS scale wherein every note of the scale can serve as a root for a 7-limit otonal or utonal tetrad, as well as either a 5-limit major or minor 7th chord. This is known as Blackwood temperament, named after Easley Blackwood, Jr., who is the first to document its existence. It has also been written on extensively by [[IgliashonJones|Igliashon Jones]] in the paper "[[http://www.cityoftheasleep.com/etc/5nEDOs.pdf|Five is Not an Odd Number]]". For an in-depth treatment of harmony in 15-edo based on this temperament (and its 7- and 11-limit extensions), see [[@Harmony in 15edo Blacksmith|Harmony in 15edo Blacksmith[10]]].

Line 643:

Line 655:

*based on treating 15-EDO as an 11-limit temperament; other approaches are possible

In <a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs">ups and downs notation</a>, which is fifth-generated, every 15edo note has at least three names. 15edo can be notated ~~without ups and downs by~~ using the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Natural%20Generators">natural generator</a>, which is not the 9\15 5th but the 2\15 2nd, also known as porcupine notation. The 15edo porcupine genchain is:

In <a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs">ups and downs notation</a>, which is fifth-generated, every 15edo note has at least three names. 15edo can also be notated using the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Natural%20Generators">natural generator</a>, which is not the 9\15 5th but the 2\15 2nd, also known as porcupine notation. The 15edo porcupine genchain is:

...A6 - A7 - A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8 - d2 - d3...

Line 654:

Line 666:

<td style="text-align: center;">cents

</td>

<td style="text-align: center;">ups and downs

<td style="text-align: center;">ups and downs relative notation

~~relative notation~~

(partial list, e.g. M2 is also A1 and d4)

</td>

<td style="text-align: center;">ups and downs

Line 700:

Line 712:

<td style="text-align: center;">160¢

</td>

<td style="text-align: center;">downaug 1, downmajor 2nd~~, downminor 3rd~~

<td style="text-align: center;">downaug unison, downmajor 2nd

</td>

<td style="text-align: center;">D#v / Ev / Fv / Gbv

Line 714:

Line 726:

<td style="text-align: center;">240¢

</td>

<td style="text-align: center;">~~aug 1,~~ maj 2nd, min 3rd~~, dim 4th~~

<td style="text-align: center;">maj 2nd, min 3rd

</td>

<td style="text-align: center;">D# / E / F / Gb

Line 728:

Line 740:

<td style="text-align: center;">320¢

</td>

<td style="text-align: center;">~~upaug 1, upmaj 2nd,~~ upmin 3rd~~, updim 4th~~

<td style="text-align: center;">upmin 3rd

</td>

<td style="text-align: center;">D#^ / E^ / F^ / Gb^

Line 742:

Line 754:

<td style="text-align: center;">400¢

</td>

<td style="text-align: center;">downmaj 3rd~~, down 4th, downdim 5th~~

<td style="text-align: center;">downmaj 3rd

</td>

<td style="text-align: center;">F#v / Gv / Abv

Line 770:

Line 782:

<td style="text-align: center;">560¢

</td>

<td style="text-align: center;">~~upmaj 3rd,~~ up 4th, updim 5th

<td style="text-align: center;">up 4th, updim 5th

</td>

<td style="text-align: center;">F#^ / G^ / Ab^

Line 784:

Line 796:

<td style="text-align: center;">640¢

</td>

<td style="text-align: center;">downaug 4th, down 5th~~, downmin 6th~~

<td style="text-align: center;">downaug 4th, down 5th

</td>

<td style="text-align: center;">G#v / Av / Bbv

Line 812:

Line 824:

<td style="text-align: center;">800¢

</td>

<td style="text-align: center;">~~upaug 4th,~~ up 5th, upmin 6th

<td style="text-align: center;">up 5th, upmin 6th

</td>

<td style="text-align: center;">G#^ / A^ / Bb^

Line 826:

Line 838:

<td style="text-align: center;">880¢

</td>

<td style="text-align: center;">downaug 5th, downmaj 6th~~, downmin 7th~~

<td style="text-align: center;">downaug 5th, downmaj 6th

</td>

<td style="text-align: center;">A#v / Bv / Cv / Dbv

Line 840:

Line 852:

<td style="text-align: center;">960¢

</td>

<td style="text-align: center;">~~aug 5th,~~ maj 6th, min 7th~~, dim 8ve~~

<td style="text-align: center;">maj 6th, min 7th

</td>

<td style="text-align: center;">A# / B / C / Db

Line 854:

Line 866:

<td style="text-align: center;">1040¢

</td>

<td style="text-align: center;">~~upmaj 6th,~~ upmin 7th~~, updim 8ve~~

<td style="text-align: center;">upmin 7th

</td>

<td style="text-align: center;">A#^ / B^ / C^ / Db^

Line 868:

Line 880:

<td style="text-align: center;">1120¢

</td>

<td style="text-align: center;">downmaj 7th, down octave~~, downmin 9th~~

<td style="text-align: center;">downmaj 7th, down octave

</td>

<td style="text-align: center;">C#v / Dv / Ebv

Line 893:

Line 905:

</table>

15edo chord names using ups and downs:

0-3-9 = D F A = Dm = &quot;D minor&quot; (or possibly D E A = Dsus2)

0-4-9 = D F^ A = D.^m = &quot;D upminor&quot;

0-5-9 = D F#v A = D.v = &quot;D dot down&quot; or &quot;D downmajor&quot;

0-6-9 = D F# A = D = &quot;D&quot; or &quot;D major&quot; (or possibly D G A = Dsus4)

0-3-9-12 = D F A C = Dm7 = &quot;D minor seven&quot;, or D F A B = Dm6 = &quot;D minor six&quot;

0-4-9-12 = D F^ A C = Dm7(^3) = &quot;D minor seven up-three&quot;, or D F^ A B = Dm6(^3) = &quot;D minor six up-three&quot;

0-5-9-12 = D F#v A C = D7(v3) = &quot;D seven down-three&quot;, or D F#v A B = D6(v3) = &quot;D six down-three&quot;

0-6-9-12 = D F# A C = D7 = &quot;D seven&quot;, or D F# A B = D6 = &quot;D six&quot;

0-5-9-14 = D F#v A C#v = D.vM7 = &quot;D downmajor seven&quot;

0-4-9-13 = D F^ A C^ = D.^m7 = &quot;D dot up minor-seven&quot;, or D F^ A B^ = D.^m6 = &quot;D dot up minor-six&quot;

For a more complete list, see <a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Chord%20names%20in%20other%20EDOs">Ups and Downs Notation - Chord names in other EDOs</a>. Because many intervals have several names, many chords do too.

15-EDO offers some minor improvements over 12-TET in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to the 7th and 11th harmonics, but its approximation to the 3rd harmonic is rather off. However, the particular way in which this approximation is off is as much a feature as it is a bug, for it allows the construction of a 5L5s MOS scale wherein every note of the scale can serve as a root for a 7-limit otonal or utonal tetrad, as well as either a 5-limit major or minor 7th chord. This is known as Blackwood temperament, named after Easley Blackwood, Jr., who is the first to document its existence. It has also been written on extensively by <a class="wiki_link" href="/IgliashonJones">Igliashon Jones</a> in the paper &quot;<a class="wiki_link_ext" href="http://www.cityoftheasleep.com/etc/5nEDOs.pdf" rel="nofollow">Five is Not an Odd Number</a>&quot;. For an in-depth treatment of harmony in 15-edo based on this temperament (and its 7- and 11-limit extensions), see <a class="wiki_link" href="/Harmony%20in%2015edo%20Blacksmith" target="_blank">Harmony in 15edo Blacksmith[10</a>].

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-11-29 01:56:47 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-11-29 02:47:08 UTC</tt>.<br>
-: The original revision id was <tt>600700440</tt>.<br>
+: The original revision id was <tt>600702192</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 75: / Line 75: @@
 *based on treating 15-EDO as an 11-limit temperament; other approaches are possible
-In [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|ups and downs notation]], which is fifth-generated, every 15edo note has at least three names. 15edo can be notated without ups and downs by using the [[Ups and Downs Notation#Natural%20Generators|natural generator]], which is not the 9\15 5th but the 2\15 2nd, also known as porcupine notation. The 15edo porcupine genchain is:
+In [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|ups and downs notation]], which is fifth-generated, every 15edo note has at least three names. 15edo can also be notated using the [[Ups and Downs Notation#Natural%20Generators|natural generator]], which is not the 9\15 5th but the 2\15 2nd, also known as porcupine notation. The 15edo porcupine genchain is:
 ...A6 - A7 - A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8 - d2 - d3...
-||= EDOstep ||= cents ||= ups and downs
+||= EDOstep ||= cents ||= ups and downs relative notation
-relative notation ||= ups and downs
+(partial list, e.g. M2 is also A1 and d4) ||= ups and downs
 absolute notation ||= porcupine
 relative notation ||= porcupine
@@ Line 85: / Line 85: @@
 ||= 0 ||= 0¢ ||= unison, min 2nd ||= C# / D / Eb ||= unison ||= D ||
 ||= 1 ||= 80¢ ||= up-unison, upminor 2nd ||= C#^ / D^ / Eb^ ||= aug unison, dim 2nd ||= D# / Eb ||
-||= 2 ||= 160¢ ||= downaug 1, downmajor 2nd, downminor 3rd ||= D#v / Ev / Fv / Gbv ||= perf 2nd ||= E ||
+||= 2 ||= 160¢ ||= downaug unison, downmajor 2nd ||= D#v / Ev / Fv / Gbv ||= perf 2nd ||= E ||
-||= 3 ||= 240¢ ||= aug 1, maj 2nd, min 3rd, dim 4th ||= D# / E / F / Gb ||= aug 2nd, dim 3rd ||= E# / Fb ||
+||= 3 ||= 240¢ ||= maj 2nd, min 3rd ||= D# / E / F / Gb ||= aug 2nd, dim 3rd ||= E# / Fb ||
-||= 4 ||= 320¢ ||= upaug 1, upmaj 2nd, upmin 3rd, updim 4th ||= D#^ / E^ / F^ / Gb^ ||= min 3rd ||= F ||
+||= 4 ||= 320¢ ||= upmin 3rd ||= D#^ / E^ / F^ / Gb^ ||= min 3rd ||= F ||
-||= 5 ||= 400¢ ||= downmaj 3rd, down 4th, downdim 5th ||= F#v / Gv / Abv ||= maj 3rd ||= F# / Gb ||
+||= 5 ||= 400¢ ||= downmaj 3rd ||= F#v / Gv / Abv ||= maj 3rd ||= F# / Gb ||
 ||= 6 ||= 480¢ ||= maj 3rd, perf 4th, dim 5th ||= F# / G / Ab ||= min 4th ||= G ||
-||= 7 ||= 560¢ ||= upmaj 3rd, up 4th, updim 5th ||= F#^ / G^ / Ab^ ||= maj 4th ||= G# / Abb ||
+||= 7 ||= 560¢ ||= up 4th, updim 5th ||= F#^ / G^ / Ab^ ||= maj 4th ||= G# / Abb ||
-||= 8 ||= 640¢ ||= downaug 4th, down 5th, downmin 6th ||= G#v / Av / Bbv ||= min 5th ||= Gx / Ab ||
+||= 8 ||= 640¢ ||= downaug 4th, down 5th ||= G#v / Av / Bbv ||= min 5th ||= Gx / Ab ||
 ||= 9 ||= 720¢ ||= aug 4th, perf 5th, min 6th ||= G# / A / Bb ||= maj 5th ||= A ||
-||= 10 ||= 800¢ ||= upaug 4th, up 5th, upmin 6th ||= G#^ / A^ / Bb^ ||= min 6th ||= A# / Bb ||
+||= 10 ||= 800¢ ||= up 5th, upmin 6th ||= G#^ / A^ / Bb^ ||= min 6th ||= A# / Bb ||
-||= 11 ||= 880¢ ||= downaug 5th, downmaj 6th, downmin 7th ||= A#v / Bv / Cv / Dbv ||= maj 6th ||= B ||
+||= 11 ||= 880¢ ||= downaug 5th, downmaj 6th ||= A#v / Bv / Cv / Dbv ||= maj 6th ||= B ||
-||= 12 ||= 960¢ ||= aug 5th, maj 6th, min 7th, dim 8ve ||= A# / B / C / Db ||= aug 6th, dim 7th ||= B# / Cb ||
+||= 12 ||= 960¢ ||= maj 6th, min 7th ||= A# / B / C / Db ||= aug 6th, dim 7th ||= B# / Cb ||
-||= 13 ||= 1040¢ ||= upmaj 6th, upmin 7th, updim 8ve ||= A#^ / B^ / C^ / Db^ ||= perf 7th ||= C ||
+||= 13 ||= 1040¢ ||= upmin 7th ||= A#^ / B^ / C^ / Db^ ||= perf 7th ||= C ||
-||= 14 ||= 1120¢ ||= downmaj 7th, down octave, downmin 9th ||= C#v / Dv / Ebv ||= aug 7th, dim 8ve ||= C# / Db ||
+||= 14 ||= 1120¢ ||= downmaj 7th, down octave ||= C#v / Dv / Ebv ||= aug 7th, dim 8ve ||= C# / Db ||
 ||= 15 ||= 1200¢ ||= maj 7th, octave, min 9th ||= C# / D / Eb ||= 8ve ||= D ||
+edo chord names using ups and downs:
+-3-9 = D F A = Dm = "D minor" (or possibly D E A = Dsus2)
+-4-9 = D F^ A = D.^m = "D upminor"
+-5-9 = D F#v A = D.v = "D dot down" or "D downmajor"
+-6-9 = D F# A = D = "D" or "D major" (or possibly D G A = Dsus4)
+-3-9-12 = D F A C = Dm7 = "D minor seven", or D F A B = Dm6 = "D minor six"
+-4-9-12 = D F^ A C = Dm7(^3) = "D minor seven up-three", or D F^ A B = Dm6(^3) = "D minor six up-three"
+-5-9-12 = D F#v A C = D7(v3) = "D seven down-three", or D F#v A B = D6(v3) = "D six down-three"
+-6-9-12 = D F# A C = D7 = "D seven", or D F# A B = D6 = "D six"
+-5-9-14 = D F#v A C#v = D.vM7 = "D downmajor seven"
+-4-9-13 = D F^ A C^ = D.^m7 = "D dot up minor-seven", or D F^ A B^ = D.^m6 = "D dot up minor-six"
+For a more complete list, see [[Ups and Downs Notation#Chord%20names%20in%20other%20EDOs|Ups and Downs Notation - Chord names in other EDOs]]. Because many intervals have several names, many chords do too.
 -EDO offers some minor improvements over 12-TET in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to the 7th and 11th harmonics, but its approximation to the 3rd harmonic is rather off. However, the particular way in which this approximation is off is as much a feature as it is a bug, for it allows the construction of a 5L5s MOS scale wherein every note of the scale can serve as a root for a 7-limit otonal or utonal tetrad, as well as either a 5-limit major or minor 7th chord. This is known as Blackwood temperament, named after Easley Blackwood, Jr., who is the first to document its existence. It has also been written on extensively by [[IgliashonJones|Igliashon Jones]] in the paper "[[http://www.cityoftheasleep.com/etc/5nEDOs.pdf|Five is Not an Odd Number]]". For an in-depth treatment of harmony in 15-edo based on this temperament (and its 7- and 11-limit extensions), see [[@Harmony in 15edo Blacksmith|Harmony in 15edo Blacksmith[10]]].
@@ Line 643: / Line 655: @@
 *based on treating 15-EDO as an 11-limit temperament; other approaches are possible&lt;br /&gt;
 &lt;br /&gt;
-In &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&gt;ups and downs notation&lt;/a&gt;, which is fifth-generated, every 15edo note has at least three names. 15edo can be notated without ups and downs by using the &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Natural%20Generators"&gt;natural generator&lt;/a&gt;, which is not the 9\15 5th but the 2\15 2nd, also known as porcupine notation. The 15edo porcupine genchain is:&lt;br /&gt;
+In &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&gt;ups and downs notation&lt;/a&gt;, which is fifth-generated, every 15edo note has at least three names. 15edo can also be notated using the &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Natural%20Generators"&gt;natural generator&lt;/a&gt;, which is not the 9\15 5th but the 2\15 2nd, also known as porcupine notation. The 15edo porcupine genchain is:&lt;br /&gt;
 ...A6 - A7 - A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8 - d2 - d3...&lt;br /&gt;
 &lt;br /&gt;
@@ Line 654: / Line 666: @@
          &lt;td style="text-align: center;"&gt;cents&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;ups and downs &lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;ups and downs relative notation&lt;br /&gt;
-relative notation&lt;br /&gt;
+(partial list, e.g. M2 is also A1 and d4)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;ups and downs&lt;br /&gt;
@@ Line 700: / Line 712: @@
          &lt;td style="text-align: center;"&gt;160¢&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;downaug 1, downmajor 2nd, downminor 3rd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;downaug unison, downmajor 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;D#v / Ev / Fv / Gbv&lt;br /&gt;
@@ Line 714: / Line 726: @@
          &lt;td style="text-align: center;"&gt;240¢&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;aug 1, maj 2nd, min 3rd, dim 4th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;maj 2nd, min 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;D# / E / F / Gb&lt;br /&gt;
@@ Line 728: / Line 740: @@
          &lt;td style="text-align: center;"&gt;320¢&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;upaug 1, upmaj 2nd, upmin 3rd, updim 4th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;upmin 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;D#^ / E^ / F^ / Gb^&lt;br /&gt;
@@ Line 742: / Line 754: @@
          &lt;td style="text-align: center;"&gt;400¢&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;downmaj 3rd, down 4th, downdim 5th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;downmaj 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;F#v / Gv / Abv&lt;br /&gt;
@@ Line 770: / Line 782: @@
          &lt;td style="text-align: center;"&gt;560¢&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;upmaj 3rd, up 4th, updim 5th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;up 4th, updim 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;F#^ / G^ / Ab^&lt;br /&gt;
@@ Line 784: / Line 796: @@
          &lt;td style="text-align: center;"&gt;640¢&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;downaug 4th, down 5th, downmin 6th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;downaug 4th, down 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;G#v / Av / Bbv&lt;br /&gt;
@@ Line 812: / Line 824: @@
          &lt;td style="text-align: center;"&gt;800¢&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;upaug 4th, up 5th, upmin 6th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;up 5th, upmin 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;G#^ / A^ / Bb^&lt;br /&gt;
@@ Line 826: / Line 838: @@
          &lt;td style="text-align: center;"&gt;880¢&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;downaug 5th, downmaj 6th, downmin 7th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;downaug 5th, downmaj 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;A#v / Bv / Cv / Dbv&lt;br /&gt;
@@ Line 840: / Line 852: @@
          &lt;td style="text-align: center;"&gt;960¢&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;aug 5th, maj 6th, min 7th, dim 8ve&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;maj 6th, min 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;A# / B / C / Db&lt;br /&gt;
@@ Line 854: / Line 866: @@
          &lt;td style="text-align: center;"&gt;1040¢&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;upmaj 6th, upmin 7th, updim 8ve&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;upmin 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;A#^ / B^ / C^ / Db^&lt;br /&gt;
@@ Line 868: / Line 880: @@
          &lt;td style="text-align: center;"&gt;1120¢&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;downmaj 7th, down octave, downmin 9th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;downmaj 7th, down octave&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C#v / Dv / Ebv&lt;br /&gt;
@@ Line 893: / Line 905: @@
 &lt;/table&gt;
+edo chord names using ups and downs:&lt;br /&gt;
+-3-9 = D F A = Dm = &amp;quot;D minor&amp;quot; (or possibly D E A = Dsus2)&lt;br /&gt;
+-4-9 = D F^ A = D.^m = &amp;quot;D upminor&amp;quot;&lt;br /&gt;
+-5-9 = D F#v A = D.v = &amp;quot;D dot down&amp;quot; or &amp;quot;D downmajor&amp;quot;&lt;br /&gt;
+-6-9 = D F# A = D = &amp;quot;D&amp;quot; or &amp;quot;D major&amp;quot; (or possibly D G A = Dsus4)&lt;br /&gt;
+-3-9-12 = D F A C = Dm7 = &amp;quot;D minor seven&amp;quot;, or D F A B = Dm6 = &amp;quot;D minor six&amp;quot;&lt;br /&gt;
+-4-9-12 = D F^ A C = Dm7(^3) = &amp;quot;D minor seven up-three&amp;quot;, or D F^ A B = Dm6(^3) = &amp;quot;D minor six up-three&amp;quot;&lt;br /&gt;
+-5-9-12 = D F#v A C = D7(v3) = &amp;quot;D seven down-three&amp;quot;, or D F#v A B = D6(v3) = &amp;quot;D six down-three&amp;quot;&lt;br /&gt;
+-6-9-12 = D F# A C = D7 = &amp;quot;D seven&amp;quot;, or D F# A B = D6 = &amp;quot;D six&amp;quot;&lt;br /&gt;
+-5-9-14 = D F#v A C#v = D.vM7 = &amp;quot;D downmajor seven&amp;quot;&lt;br /&gt;
+-4-9-13 = D F^ A C^ = D.^m7 = &amp;quot;D dot up minor-seven&amp;quot;, or D F^ A B^ = D.^m6 = &amp;quot;D dot up minor-six&amp;quot;&lt;br /&gt;
+For a more complete list, see &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Chord%20names%20in%20other%20EDOs"&gt;Ups and Downs Notation - Chord names in other EDOs&lt;/a&gt;. Because many intervals have several names, many chords do too.&lt;br /&gt;
 &lt;br /&gt;
 -EDO offers some minor improvements over 12-TET in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to the 7th and 11th harmonics, but its approximation to the 3rd harmonic is rather off. However, the particular way in which this approximation is off is as much a feature as it is a bug, for it allows the construction of a 5L5s MOS scale wherein every note of the scale can serve as a root for a 7-limit otonal or utonal tetrad, as well as either a 5-limit major or minor 7th chord. This is known as Blackwood temperament, named after Easley Blackwood, Jr., who is the first to document its existence. It has also been written on extensively by &lt;a class="wiki_link" href="/IgliashonJones"&gt;Igliashon Jones&lt;/a&gt; in the paper &amp;quot;&lt;a class="wiki_link_ext" href="http://www.cityoftheasleep.com/etc/5nEDOs.pdf" rel="nofollow"&gt;Five is Not an Odd Number&lt;/a&gt;&amp;quot;. For an in-depth treatment of harmony in 15-edo based on this temperament (and its 7- and 11-limit extensions), see &lt;a class="wiki_link" href="/Harmony%20in%2015edo%20Blacksmith" target="_blank"&gt;Harmony in 15edo Blacksmith[10&lt;/a&gt;].&lt;br /&gt;