21edo: Difference between revisions

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<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:~~JosephRuhf~~|~~JosephRuhf~~]] and made on <tt>2016-12-04 23:~~21:03~~ UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-07 21:23:55 UTC</tt>.

: The original revision id was <tt>~~601376472~~</tt>.

: The original revision id was <tt>601671926</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.

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The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit [[Laconic Family|laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic, spartan and gorgo temperaments. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.

|| **Degree** || **Cents coarse/fine, DMS**

|| **Degree** || **Cents coarse/fine, DMS Value** ||||= [[Ups and Downs Notation|Up/down]]

**Value** ||||= [[Ups and Downs Notation|Up/down]]

[[Ups and Downs Notation|Notation]] || **5L3s**

**Notation** ||= **D.-R. Interval**

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|| 0 || 0 ||= C ||= unison || C ||= Unison ||= 1/1 ||= 1/1 ||= 1/1 ||

|| 1 || 57.143, 68.571

17°8'34" ||= C^/Dvv ||= up unison || C# ||= Subminor 2nd ||= 28/27, 30/29 ||= 35/34, 36/35 ||= 64/63 ||

17°8'34" ||= C^/Dvv ||= up unison,

double-down

2nd || C# ||= Subminor 2nd ||= 28/27, 30/29 ||= 35/34, 36/35 ||= 64/63 ||

|| 2 || 114.286, 137.143

34°17'9" ||= C^^/Dv ||= down 2nd || Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||= 18/17 ||= 16/15, 25/24 ||

34°17'9" ||= C^^/Dv ||= double-up

unison,

down 2nd || Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||= 18/17 ||= 16/15, 25/24 ||

|| 3 || 171.429, 205.714

51°25'43" ||= D ||= 2nd || D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 ||

51°25'43" ||= D ||= perfect 2nd || D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 ||

|| 4 || 228.571, 274.286

68°34'17" ||= D^/Evv ||= up 2nd || D# ||= Supermajor 2nd ||= 8/7 ||= 8/7 ||= 8/7, 10/9, 11/10 ||

68°34'17" ||= D^/Evv ||= up 2nd,

double-down 3rd || D# ||= Supermajor 2nd ||= 8/7 ||= 8/7 ||= 8/7, 10/9, 11/10 ||

|| 5 || 285.714, 342.857

85°42'51" ||= D^^/Ev ||= down 3rd || Eb ||= Subminor 3rd ||= 27/23, 32/27 ||= 13/11, 20/17 ||= 6/5, 7/6 ||

85°42'51" ||= D^^/Ev ||= double-up

2nd,

down 3rd || Eb ||= Subminor 3rd ||= 27/23, 32/27 ||= 13/11, 20/17 ||= 6/5, 7/6 ||

|| 6 || 342.857, 411.429

102°51'26" ||= E ||= 3rd || E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 ||

102°51'26" ||= E ||= perfect 3rd || E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 ||

|| 7 || 400, 480

120° ||= E^/Fvv ||= up 3rd || E#/Fb ||= Major 3rd ||= 29/23 ||= 44/35 ||= 5/4, 9/7, 11/9, 14/11 ||

120° ||= E^/Fvv ||= up 3rd,

double-down

4th || E#/Fb ||= Major 3rd ||= 29/23 ||= 44/35 ||= 5/4, 9/7, 11/9, 14/11 ||

|| 8 || 457.143, 548.571

137°8'34" ||= E^^/Fv ||= down 4th || F ||= Third-Fourth ||= 30/23 ||= 13/10, 17/13, 22/17 ||= 13/10 ||

137°8'34" ||= E^^/Fv ||= double-up

3rd,

down 4th || F ||= Third-Fourth ||= 30/23 ||= 13/10, 17/13, 22/17 ||= 13/10 ||

|| 9 || 514.286, 617.143

154°17'9" ||= F ||= 4th || F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 ||

154°17'9" ||= F ||= perfect 4th || F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 ||

|| 10 || 571.429, 685.714

171°25'43" ||= F^/Gvv ||= up 4th || Gb ||= Narrow Tritone ||= 32/23 ||= 18/13 ||= 7/5, 11/8 ||

171°25'43" ||= F^/Gvv ||= up 4th,

double-down

5th || Gb ||= Narrow Tritone ||= 32/23 ||= 18/13 ||= 7/5, 11/8 ||

|| 11 || 628.571, 754.286

188°34'17" ||= F^^/Gv ||= down 5th || G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 ||

188°34'17" ||= F^^/Gv ||= double-up

4th,

down 5th || G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 ||

|| 12 || 685.714, 824.857

205°42'51" ||= G ||= 5th || G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 ||

205°42'51" ||= G ||= perfect 5th || G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 ||

|| 13 || 742.857, 891.428

222°51'26" ||= G^/Avv ||= up 5th || Hb ||= Fifth-Sixth ||= 23/15 ||= 17/11, 20/13, 26/17 ||= 20/13 ||

222°51'26" ||= G^/Avv ||= up 5th,

double-down

6th || Hb ||= Fifth-Sixth ||= 23/15 ||= 17/11, 20/13, 26/17 ||= 20/13 ||

|| 14 || 800, 960

240° ||= G^^/Av ||= down 6th || H ||= Minor 6th ||= 46/29 ||= 35/22 ||= 8/5, 11/7, 14/9, 18/11 ||

240° ||= G^^/Av ||= double-up

5th, down 6th || H ||= Minor 6th ||= 46/29 ||= 35/22 ||= 8/5, 11/7, 14/9, 18/11 ||

|| 15 || 857.143, 1028.571

257°8'34" ||= A ||= 6th || H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 ||

257°8'34" ||= A ||= perfect 6th || H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 ||

|| 16 || 914.286, 1097.143

274°17'9" ||= A^/Bvv ||= up 6th || A ||= Supermajor 6th ||= 27/16, 46/27 ||= 17/10, 22/13 ||= 5/3, 12/7 ||

274°17'9" ||= A^/Bvv ||= up 6th,

double-down

7th || A ||= Supermajor 6th ||= 27/16, 46/27 ||= 17/10, 22/13 ||= 5/3, 12/7 ||

|| 17 || 971.429, 1165.714

391°25'43" ||= A^^/Bv ||= down 7th || A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 ||

391°25'43" ||= A^^/Bv ||= double-up

6th,

down 7th || A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 ||

|| 18 || 1028.571, 1234.286

308°34'17" ||= B ||= 7th || Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 ||

308°34'17" ||= B ||= perfect 7th || Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 ||

|| 19 || 1085.714, 1302.857

325°42'51" ||= B^/Cvv ||= up 7th || B ||= Major 7th ||= 15/8 ||= 17/9 ||= 15/8, 48/25 ||

325°42'51" ||= B^/Cvv ||= up 7th,

double-down

8ve || B ||= Major 7th ||= 15/8 ||= 17/9 ||= 15/8, 48/25 ||

|| 20 || 1142.857, 1371.429

342°8'34" ||= B^^/Cv ||= down 8ve || B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||= 35/18, 68/35 ||= 63/32 ||

342°8'34" ||= B^^/Cv ||= double-up

7th, down 8ve || B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||= 35/18, 68/35 ||= 63/32 ||

|| 21 || 1200, 1440

360° ||= C ||= 8ve || C ||= Octave ||= 2/1 ||= 2/1 ||= 2/1 ||

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==Triadic Harmony in 21-EDO:==

One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as "3rds" for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds (~~respectively~~). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy "altered" triads that stand out as representations to parts of the overtone series:

One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as "3rds" for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds respectively (or double-down, down, perfect, up and double-up). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy "altered" triads that stand out as representations to parts of the overtone series:

||= **Steps** ||= **Cents** ||= **Ratio** ||

||= **Steps** ||= **Cents** ||= **Ratio** || **Example in C** || **Written name** || **Spoken name** ||

||= 0-5-10 ||= 0-286-571 ||= 23:27:32 ||

||= 0-5-10 ||= 0-286-571 ||= 23:27:32 || C Ev Gvv || C.v(vv5) || C dot down, double-down five ||

||= 0-4-11 ||= 0-229-629 ||= 7:8:10 ||

||= 0-4-11 ||= 0-229-629 ||= 7:8:10 || C Evv Gv || C.vv(v5) || C dot double-down, down five ||

||= 0-6-11 ||= 0-343-629 ||= 9:11:13 ||

||= 0-6-11 ||= 0-343-629 ||= 9:11:13 || C E Gv || C(v5) || C down-five ||

||= 0-5-13 ||= 0-286-743 ||= 11:13:17 ||

||= 0-5-13 ||= 0-286-743 ||= 11:13:17 || C Ev G^ || C.v(^5) || C dot down up-five ||

||= 0-8-13 ||= 0-457-743 ||= 13:17:20 ||

||= 0-8-13 ||= 0-457-743 ||= 13:17:20 || C Fv G^ || C.v4(^5) || C (sus) down-four up-five ||

||= 0-5-15 ||= 0-286-857 ||= 11:13:18 ||

||= 0-5-15 ||= 0-286-857 ||= 11:13:18 || C Ev A || A(v5) || (inversion of 9:11:13) ||

==Moment-of-Symmetry Scales in 21-EDO:==

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While 21-EDO lacks any 7-note MOS scales, one can still construct a variety of interesting and useful 7-note scales using tetrachords instead of MOS generators. The 21-EDO fourth is 9 steps, which can be divided into three parts in the following ways:

||= Step Pattern ||= Cents ||= Name* ||

||= Step Pattern ||= Cents || Example ||= Name* ||

||= 3, 3, 3 ||= (0)-171-343-(514) ||= Equable diatonic ||

||= 3, 3, 3 ||= (0)-171-343-(514) || C D E F ||= Equable diatonic ||

||= 4, 3, 2 ||= (0)-229-400-(514) ||= Soft diatonic ||

||= 4, 3, 2 ||= (0)-229-400-(514) || C D^ E^ F ||= Soft diatonic ||

||= 4, 4, 1 ||= (0)-229-457-(514) ||= Intense diatonic ||

||= 4, 4, 1 ||= (0)-229-457-(514) || C D^ E^^ F ||= Intense diatonic ||

||= 5, 3, 1 ||= (0)-286-457-(514) ||= Archytas chromatic ||

||= 5, 3, 1 ||= (0)-286-457-(514) || C D^^ E^^ F ||= Archytas chromatic ||

||= 5, 2, 2 ||= (0)-286-400-(514) ||= Weak chromatic ||

||= 5, 2, 2 ||= (0)-286-400-(514) || C D^^ E^ F ||= Weak chromatic ||

||= 6, 2, 1 ||= (0)-343-457-(514) ||= Strong enharmonic ||

||= 6, 2, 1 ||= (0)-343-457-(514) || C D^3 E^^ F ||= Strong enharmonic ||

||= 7, 1, 1 ||= (0)-400-457-(514) ||= Pythagorean enharmonic ||

||= 7, 1, 1 ||= (0)-400-457-(514) || C D^4 E^^ F ||= Pythagorean enharmonic ||

*these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!

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<td>Degree

</td>

<td>Cents coarse/fine, DMS~~ ~~

<td>Cents coarse/fine, DMS Value

~~~~Value

</td>

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<td style="text-align: center;">C^/Dvv

</td>

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

<td style="text-align: center;">up unison,

double-down

2nd

</td>

<td>C#

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<td style="text-align: center;">C^^/Dv

</td>

<td style="text-align: center;">down 2nd

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

unison,

down 2nd

</td>

<td>Db

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

</td>

<td style="text-align: center;">2nd

<td style="text-align: center;">perfect 2nd

</td>

<td>D

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<td style="text-align: center;">D^/Evv

</td>

<td style="text-align: center;">up 2nd

<td style="text-align: center;">up 2nd,

double-down 3rd

</td>

<td>D#

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<td style="text-align: center;">D^^/Ev

</td>

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

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

2nd,

down 3rd

</td>

<td>Eb

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

</td>

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

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

</td>

<td>E

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<td style="text-align: center;">E^/Fvv

</td>

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

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

double-down

4th

</td>

<td>E#/Fb

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<td style="text-align: center;">E^^/Fv

</td>

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

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

3rd,

down 4th

</td>

<td>F

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

</td>

<td style="text-align: center;">4th

<td style="text-align: center;">perfect 4th

</td>

<td>F#

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<td style="text-align: center;">F^/Gvv

</td>

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

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

double-down

5th

</td>

<td>Gb

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<td style="text-align: center;">F^^/Gv

</td>

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

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

4th,

down 5th

</td>

<td>G

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

</td>

<td style="text-align: center;">5th

<td style="text-align: center;">perfect 5th

</td>

<td>G#

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<td style="text-align: center;">G^/Avv

</td>

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

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

double-down

6th

</td>

<td>Hb

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<td style="text-align: center;">G^^/Av

</td>

<td style="text-align: center;">down 6th

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

5th, down 6th

</td>

<td>H

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

</td>

<td style="text-align: center;">6th

<td style="text-align: center;">perfect 6th

</td>

<td>H#/Ab

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<td style="text-align: center;">A^/Bvv

</td>

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

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

double-down

7th

</td>

<td>A

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<td style="text-align: center;">A^^/Bv

</td>

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

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

6th,

down 7th

</td>

<td>A#

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

</td>

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

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

</td>

<td>Bb

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<td style="text-align: center;">B^/Cvv

</td>

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

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

double-down

8ve

</td>

<td>B

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<td style="text-align: center;">B^^/Cv

</td>

<td style="text-align: center;">down 8ve

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

7th, down 8ve

</td>

<td>B#/Cb

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<h2 id="toc2"><a name="x21 equal divisions of the octave-Triadic Harmony in 21-EDO:"></a>Triadic Harmony in 21-EDO:</h2>

One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as &quot;3rds&quot; for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds (~~respectively~~). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy &quot;altered&quot; triads that stand out as representations to parts of the overtone series:

One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as &quot;3rds&quot; for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds respectively (or double-down, down, perfect, up and double-up). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy &quot;altered&quot; triads that stand out as representations to parts of the overtone series:

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</td>

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

</td>

<td>Example in C

</td>

<td>Written name

</td>

<td>Spoken name

</td>

</tr>

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</td>

<td style="text-align: center;">23:27:32

</td>

<td>C Ev Gvv

</td>

<td>C.v(vv5)

</td>

<td>C dot down, double-down five

</td>

</tr>

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</td>

<td style="text-align: center;">7:8:10

</td>

<td>C Evv Gv

</td>

<td>C.vv(v5)

</td>

<td>C dot double-down, down five

</td>

</tr>

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</td>

<td style="text-align: center;">9:11:13

</td>

<td>C E Gv

</td>

<td>C(v5)

</td>

<td>C down-five

</td>

</tr>

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</td>

<td style="text-align: center;">11:13:17

</td>

<td>C Ev G^

</td>

<td>C.v(^5)

</td>

<td>C dot down up-five

</td>

</tr>

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</td>

<td style="text-align: center;">13:17:20

</td>

<td>C Fv G^

</td>

<td>C.v4(^5)

</td>

<td>C (sus) down-four up-five

</td>

</tr>

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</td>

<td style="text-align: center;">11:13:18

</td>

<td>C Ev A

</td>

<td>A(v5)

</td>

<td>(inversion of 9:11:13)

</td>

</tr>

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</td>

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

</td>

<td>Example

</td>

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

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</td>

<td style="text-align: center;">(0)-171-343-(514)

</td>

<td>C D E F

</td>

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

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</td>

<td style="text-align: center;">(0)-229-400-(514)

</td>

<td>C D^ E^ F

</td>

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

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</td>

<td style="text-align: center;">(0)-229-457-(514)

</td>

<td>C D^ E^^ F

</td>

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

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</td>

<td style="text-align: center;">(0)-286-457-(514)

</td>

<td>C D^^ E^^ F

</td>

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

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</td>

<td style="text-align: center;">(0)-286-400-(514)

</td>

<td>C D^^ E^ F

</td>

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

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</td>

<td style="text-align: center;">(0)-343-457-(514)

</td>

<td>C D^3 E^^ F

</td>

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

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</td>

<td style="text-align: center;">(0)-400-457-(514)

</td>

<td>C D^4 E^^ F

</td>

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

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<h1 id="toc7"><a name="Books / Literature:"></a>Books / Literature:</h1>

Sword, Ron. &quot;Icosihenaphonic Scales for Guitar&quot;. IAAA Press. 1st ed: July 2009.

<img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /><img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" />

<img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /><img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" />

21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)

@@ 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:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-04 23:21:03 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-07 21:23:55 UTC</tt>.<br>
-: The original revision id was <tt>601376472</tt>.<br>
+: The original revision id was <tt>601671926</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 19: / Line 19: @@
 The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit [[Laconic Family|laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic, spartan and gorgo temperaments. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.
-|| **Degree** || **Cents coarse/fine, DMS**
+|| **Degree** || **Cents coarse/fine, DMS Value** ||||= [[Ups and Downs Notation|Up/down]]
-**Value** ||||= [[Ups and Downs Notation|Up/down]]
 [[Ups and Downs Notation|Notation]] || **5L3s**
 **Notation** ||= **D.-R. Interval**
@@ Line 27: / Line 26: @@
 || 0 || 0 ||= C ||= unison || C ||= Unison ||= 1/1 ||= 1/1 ||= 1/1 ||
 || 1 || 57.143, 68.571
-°&lt;span style="background-color: #ffffff;"&gt;8'34"&lt;/span&gt; ||= C^/Dvv ||= up unison || C# ||= Subminor 2nd ||= 28/27, 30/29 ||= 35/34, 36/35 ||= 64/63 ||
+°&lt;span style="background-color: #ffffff;"&gt;8'34"&lt;/span&gt; ||= C^/Dvv ||= up unison,
+double-down
+nd || C# ||= Subminor 2nd ||= 28/27, 30/29 ||= 35/34, 36/35 ||= 64/63 ||
 || 2 || 114.286, 137.143
-°&lt;span style="background-color: #ffffff;"&gt;17'9"&lt;/span&gt; ||= C^^/Dv ||= down 2nd || Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||= 18/17 ||= 16/15, 25/24 ||
+°&lt;span style="background-color: #ffffff;"&gt;17'9"&lt;/span&gt; ||= C^^/Dv ||= double-up
+unison,
+down 2nd || Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||= 18/17 ||= 16/15, 25/24 ||
 || 3 || 171.429, 205.714
-&lt;span style="background-color: #ffffff;"&gt;51°25'43"&lt;/span&gt; ||= D ||= 2nd || D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 ||
+&lt;span style="background-color: #ffffff;"&gt;51°25'43"&lt;/span&gt; ||= D ||= perfect 2nd || D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 ||
 || 4 || 228.571, 274.286
-°&lt;span style="background-color: #ffffff;"&gt;34'17"&lt;/span&gt; ||= D^/Evv ||= up 2nd || D# ||= Supermajor 2nd ||= 8/7 ||= 8/7 ||= 8/7, 10/9, 11/10 ||
+°&lt;span style="background-color: #ffffff;"&gt;34'17"&lt;/span&gt; ||= D^/Evv ||= up 2nd,
+double-down 3rd || D# ||= Supermajor 2nd ||= 8/7 ||= 8/7 ||= 8/7, 10/9, 11/10 ||
 || 5 || 285.714, 342.857
-°&lt;span style="background-color: #ffffff;"&gt;42'51"&lt;/span&gt; ||= D^^/Ev ||= down 3rd || Eb ||= Subminor 3rd ||= 27/23, 32/27 ||= 13/11, 20/17 ||= 6/5, 7/6 ||
+°&lt;span style="background-color: #ffffff;"&gt;42'51"&lt;/span&gt; ||= D^^/Ev ||= double-up
+nd,
+down 3rd || Eb ||= Subminor 3rd ||= 27/23, 32/27 ||= 13/11, 20/17 ||= 6/5, 7/6 ||
 || 6 || 342.857, 411.429
-&lt;span style="background-color: #ffffff;"&gt;102°51'26"&lt;/span&gt; ||= E ||= 3rd || E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 ||
+&lt;span style="background-color: #ffffff;"&gt;102°51'26"&lt;/span&gt; ||= E ||= perfect 3rd || E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 ||
 || 7 || 400, 480
-° ||= E^/Fvv ||= up 3rd || E#/Fb ||= Major 3rd ||= 29/23 ||= 44/35 ||= 5/4, 9/7, 11/9, 14/11 ||
+° ||= E^/Fvv ||= up 3rd,
+double-down
+th || E#/Fb ||= Major 3rd ||= 29/23 ||= 44/35 ||= 5/4, 9/7, 11/9, 14/11 ||
 || 8 || 457.143, 548.571
-°&lt;span style="background-color: #ffffff; line-height: 1.5;"&gt;8'34"&lt;/span&gt; ||= E^^/Fv ||= down 4th || F ||= Third-Fourth ||= 30/23 ||= 13/10, 17/13, 22/17 ||= 13/10 ||
+°&lt;span style="background-color: #ffffff; line-height: 1.5;"&gt;8'34"&lt;/span&gt; ||= E^^/Fv ||= double-up
+rd,
+down 4th || F ||= Third-Fourth ||= 30/23 ||= 13/10, 17/13, 22/17 ||= 13/10 ||
 || 9 || 514.286, 617.143
-&lt;span style="background-color: #ffffff;"&gt;154°17'9"&lt;/span&gt; ||= F ||= 4th || F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 ||
+&lt;span style="background-color: #ffffff;"&gt;154°17'9"&lt;/span&gt; ||= F ||= perfect 4th || F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 ||
 || 10 || 571.429, 685.714
-°&lt;span style="background-color: #ffffff;"&gt;25'43"&lt;/span&gt; ||= F^/Gvv ||= up 4th || Gb ||= Narrow Tritone ||= 32/23 ||= 18/13 ||= 7/5, 11/8 ||
+°&lt;span style="background-color: #ffffff;"&gt;25'43"&lt;/span&gt; ||= F^/Gvv ||= up 4th,
+double-down
+th || Gb ||= Narrow Tritone ||= 32/23 ||= 18/13 ||= 7/5, 11/8 ||
 || 11 || 628.571, 754.286
-°&lt;span style="background-color: #ffffff;"&gt;34'17"&lt;/span&gt; ||= F^^/Gv ||= down 5th || G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 ||
+°&lt;span style="background-color: #ffffff;"&gt;34'17"&lt;/span&gt; ||= F^^/Gv ||= double-up
+th,
+down 5th || G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 ||
 || 12 || 685.714, 824.857
-&lt;span style="background-color: #ffffff;"&gt;205°42'51"&lt;/span&gt; ||= G ||= 5th || G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 ||
+&lt;span style="background-color: #ffffff;"&gt;205°42'51"&lt;/span&gt; ||= G ||= perfect 5th || G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 ||
 || 13 || 742.857, 891.428
-°&lt;span style="background-color: #ffffff;"&gt;51'26"&lt;/span&gt; ||= G^/Avv ||= up 5th || Hb ||= Fifth-Sixth ||= 23/15 ||= 17/11, 20/13, 26/17 ||= 20/13 ||
+°&lt;span style="background-color: #ffffff;"&gt;51'26"&lt;/span&gt; ||= G^/Avv ||= up 5th,
+double-down
+th || Hb ||= Fifth-Sixth ||= 23/15 ||= 17/11, 20/13, 26/17 ||= 20/13 ||
 || 14 || 800, 960
-° ||= G^^/Av ||= down 6th || H ||= Minor 6th ||= 46/29 ||= 35/22 ||= 8/5, 11/7, 14/9, 18/11 ||
+° ||= G^^/Av ||= double-up
+th, down 6th || H ||= Minor 6th ||= 46/29 ||= 35/22 ||= 8/5, 11/7, 14/9, 18/11 ||
 || 15 || 857.143, 1028.571
-&lt;span style="background-color: #ffffff;"&gt;257°8'34"&lt;/span&gt; ||= A ||= 6th || H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 ||
+&lt;span style="background-color: #ffffff;"&gt;257°8'34"&lt;/span&gt; ||= A ||= perfect 6th || H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 ||
 || 16 || 914.286, 1097.143
-°&lt;span style="background-color: #ffffff;"&gt;17'9"&lt;/span&gt; ||= A^/Bvv ||= up 6th || A ||= Supermajor 6th ||= 27/16, 46/27 ||= 17/10, 22/13 ||= 5/3, 12/7 ||
+°&lt;span style="background-color: #ffffff;"&gt;17'9"&lt;/span&gt; ||= A^/Bvv ||= up 6th,
+double-down
+th || A ||= Supermajor 6th ||= 27/16, 46/27 ||= 17/10, 22/13 ||= 5/3, 12/7 ||
 || 17 || 971.429, 1165.714
-°&lt;span style="background-color: #ffffff;"&gt;25'43"&lt;/span&gt; ||= A^^/Bv ||= down 7th || A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 ||
+°&lt;span style="background-color: #ffffff;"&gt;25'43"&lt;/span&gt; ||= A^^/Bv ||= double-up
+th,
+down 7th || A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 ||
 || 18 || 1028.571, 1234.286
-&lt;span style="background-color: #ffffff;"&gt;308°34'17"&lt;/span&gt; ||= B ||= 7th || Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 ||
+&lt;span style="background-color: #ffffff;"&gt;308°34'17"&lt;/span&gt; ||= B ||= perfect 7th || Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 ||
 || 19 || 1085.714, 1302.857
-°&lt;span style="background-color: #ffffff;"&gt;42'51"&lt;/span&gt; ||= B^/Cvv ||= up 7th || B ||= Major 7th ||= 15/8 ||= 17/9 ||= 15/8, 48/25 ||
+°&lt;span style="background-color: #ffffff;"&gt;42'51"&lt;/span&gt; ||= B^/Cvv ||= up 7th,
+double-down
+ve || B ||= Major 7th ||= 15/8 ||= 17/9 ||= 15/8, 48/25 ||
 || 20 || 1142.857, 1371.429
-°&lt;span style="background-color: #ffffff;"&gt;8'34"&lt;/span&gt; ||= B^^/Cv ||= down 8ve || B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||= 35/18, 68/35 ||= 63/32 ||
+°&lt;span style="background-color: #ffffff;"&gt;8'34"&lt;/span&gt; ||= B^^/Cv ||= double-up
+th, down 8ve || B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||= 35/18, 68/35 ||= 63/32 ||
 || 21 || 1200, 1440
 ° ||= C ||= 8ve || C ||= Octave ||= 2/1 ||= 2/1 ||= 2/1 ||
@@ Line 80: / Line 104: @@
 ==Triadic Harmony in 21-EDO:==
-One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as "3rds" for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds (respectively). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy "altered" triads that stand out as representations to parts of the overtone series:
+One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as "3rds" for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds respectively (or double-down, down, perfect, up and double-up). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy "altered" triads that stand out as representations to parts of the overtone series:
-||= **Steps** ||= **Cents** ||= **Ratio** ||
+||= **Steps** ||= **Cents** ||= **Ratio** || **Example in C** || **Written name** || **Spoken name** ||
-||= 0-5-10 ||= 0-286-571 ||= 23:27:32 ||
+||= 0-5-10 ||= 0-286-571 ||= 23:27:32 || C Ev Gvv || C.v(vv5) || C dot down, double-down five ||
-||= 0-4-11 ||= 0-229-629 ||= 7:8:10 ||
+||= 0-4-11 ||= 0-229-629 ||= 7:8:10 || C Evv Gv || C.vv(v5) || C dot double-down, down five ||
-||= 0-6-11 ||= 0-343-629 ||= 9:11:13 ||
+||= 0-6-11 ||= 0-343-629 ||= 9:11:13 || C E Gv || C(v5) || C down-five ||
-||= 0-5-13 ||= 0-286-743 ||= 11:13:17 ||
+||= 0-5-13 ||= 0-286-743 ||= 11:13:17 || C Ev G^ || C.v(^5) || C dot down up-five ||
-||= 0-8-13 ||= 0-457-743 ||= 13:17:20 ||
+||= 0-8-13 ||= 0-457-743 ||= 13:17:20 || C Fv G^ || C.v4(^5) || C (sus) down-four up-five ||
-||= 0-5-15 ||= 0-286-857 ||= 11:13:18 ||
+||= 0-5-15 ||= 0-286-857 ||= 11:13:18 || C Ev A || A(v5) || (inversion of 9:11:13) ||
 ==Moment-of-Symmetry Scales in 21-EDO:==
@@ Line 100: / Line 124: @@
 While 21-EDO lacks any 7-note MOS scales, one can still construct a variety of interesting and useful 7-note scales using tetrachords instead of MOS generators. The 21-EDO fourth is 9 steps, which can be divided into three parts in the following ways:
-||= Step Pattern ||= Cents ||= Name* ||
+||= Step Pattern ||= Cents || Example ||= Name* ||
-||= 3, 3, 3 ||= (0)-171-343-(514) ||= Equable diatonic ||
+||= 3, 3, 3 ||= (0)-171-343-(514) || C D E F ||= Equable diatonic ||
-||= 4, 3, 2 ||= (0)-229-400-(514) ||= Soft diatonic ||
+||= 4, 3, 2 ||= (0)-229-400-(514) || C D^ E^ F ||= Soft diatonic ||
-||= 4, 4, 1 ||= (0)-229-457-(514) ||= Intense diatonic ||
+||= 4, 4, 1 ||= (0)-229-457-(514) || C D^ E^^ F ||= Intense diatonic ||
-||= 5, 3, 1 ||= (0)-286-457-(514) ||= Archytas chromatic ||
+||= 5, 3, 1 ||= (0)-286-457-(514) || C D^^ E^^ F ||= Archytas chromatic ||
-||= 5, 2, 2 ||= (0)-286-400-(514) ||= Weak chromatic ||
+||= 5, 2, 2 ||= (0)-286-400-(514) || C D^^ E^ F ||= Weak chromatic ||
-||= 6, 2, 1 ||= (0)-343-457-(514) ||= Strong enharmonic ||
+||= 6, 2, 1 ||= (0)-343-457-(514) || C D^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; E^^ F ||= Strong enharmonic ||
-||= 7, 1, 1 ||= (0)-400-457-(514) ||= Pythagorean enharmonic ||
+||= 7, 1, 1 ||= (0)-400-457-(514) || C D^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; E^^ F ||= Pythagorean enharmonic ||
 *these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!
@@ Line 183: / Line 207: @@
          &lt;td&gt;&lt;strong&gt;Degree&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;Cents coarse/fine, DMS&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;Cents coarse/fine, DMS Value&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;Value&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td colspan="2" style="text-align: center;"&gt;&lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;Up/down&lt;/a&gt;&lt;br /&gt;
@@ Line 231: / Line 254: @@
          &lt;td style="text-align: center;"&gt;C^/Dvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;up unison&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;up unison,&lt;br /&gt;
+double-down&lt;br /&gt;
+nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;C#&lt;br /&gt;
@@ Line 252: / Line 277: @@
          &lt;td style="text-align: center;"&gt;C^^/Dv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;down 2nd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;double-up&lt;br /&gt;
+unison,&lt;br /&gt;
+down 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Db&lt;br /&gt;
@@ Line 273: / Line 300: @@
          &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;2nd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;perfect 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;D&lt;br /&gt;
@@ Line 294: / Line 321: @@
          &lt;td style="text-align: center;"&gt;D^/Evv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;up 2nd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;up 2nd,&lt;br /&gt;
+double-down 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;D#&lt;br /&gt;
@@ Line 315: / Line 343: @@
          &lt;td style="text-align: center;"&gt;D^^/Ev&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;down 3rd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;double-up&lt;br /&gt;
+nd, &lt;br /&gt;
+down 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Eb&lt;br /&gt;
@@ Line 336: / Line 366: @@
          &lt;td style="text-align: center;"&gt;E&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;3rd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;perfect 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;E&lt;br /&gt;
@@ Line 357: / Line 387: @@
          &lt;td style="text-align: center;"&gt;E^/Fvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;up 3rd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;up 3rd,&lt;br /&gt;
+double-down&lt;br /&gt;
+th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;E#/Fb&lt;br /&gt;
@@ Line 378: / Line 410: @@
          &lt;td style="text-align: center;"&gt;E^^/Fv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;down 4th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;double-up&lt;br /&gt;
+rd,&lt;br /&gt;
+down 4th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;F&lt;br /&gt;
@@ Line 399: / Line 433: @@
          &lt;td style="text-align: center;"&gt;F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;4th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;perfect 4th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;F#&lt;br /&gt;
@@ Line 420: / Line 454: @@
          &lt;td style="text-align: center;"&gt;F^/Gvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;up 4th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;up 4th,&lt;br /&gt;
+double-down&lt;br /&gt;
+th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Gb&lt;br /&gt;
@@ Line 441: / Line 477: @@
          &lt;td style="text-align: center;"&gt;F^^/Gv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;down 5th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;double-up&lt;br /&gt;
+th,&lt;br /&gt;
+down 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;G&lt;br /&gt;
@@ Line 462: / Line 500: @@
          &lt;td style="text-align: center;"&gt;G&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;5th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;perfect 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;G#&lt;br /&gt;
@@ Line 483: / Line 521: @@
          &lt;td style="text-align: center;"&gt;G^/Avv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;up 5th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;up 5th,&lt;br /&gt;
+double-down&lt;br /&gt;
+th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Hb&lt;br /&gt;
@@ Line 504: / Line 544: @@
          &lt;td style="text-align: center;"&gt;G^^/Av&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;down 6th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;double-up&lt;br /&gt;
+th, down 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;H&lt;br /&gt;
@@ Line 525: / Line 566: @@
          &lt;td style="text-align: center;"&gt;A&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;6th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;perfect 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;H#/Ab&lt;br /&gt;
@@ Line 546: / Line 587: @@
          &lt;td style="text-align: center;"&gt;A^/Bvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;up 6th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;up 6th,&lt;br /&gt;
+double-down&lt;br /&gt;
+th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;A&lt;br /&gt;
@@ Line 567: / Line 610: @@
          &lt;td style="text-align: center;"&gt;A^^/Bv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;down 7th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;double-up&lt;br /&gt;
+th,&lt;br /&gt;
+down 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;A#&lt;br /&gt;
@@ Line 588: / Line 633: @@
          &lt;td style="text-align: center;"&gt;B&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;7th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;perfect 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Bb&lt;br /&gt;
@@ Line 609: / Line 654: @@
          &lt;td style="text-align: center;"&gt;B^/Cvv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;up 7th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;up 7th,&lt;br /&gt;
+double-down&lt;br /&gt;
+ve&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;B&lt;br /&gt;
@@ Line 630: / Line 677: @@
          &lt;td style="text-align: center;"&gt;B^^/Cv&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;down 8ve&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;double-up&lt;br /&gt;
+th, down 8ve&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;B#/Cb&lt;br /&gt;
@@ Line 678: / Line 726: @@
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x21 equal divisions of the octave-Triadic Harmony in 21-EDO:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Triadic Harmony in 21-EDO:&lt;/h2&gt;
   &lt;br /&gt;
-One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as &amp;quot;3rds&amp;quot; for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds (respectively). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy &amp;quot;altered&amp;quot; triads that stand out as representations to parts of the overtone series:&lt;br /&gt;
+One interesting feature of 21-EDO is the variety of triads it offers. Five of its intervals--228.6¢, 285.7¢, 342.9¢, 400¢, and 457.1¢ can function categorically as &amp;quot;3rds&amp;quot; for those whose ears are accustomed to diatonic interval categories, representing arto, minor, neutral, major, and tendo 3rds respectively (or double-down, down, perfect, up and double-up). One can couple these with 21-EDO's narrow fifth to form five types of triad. In addition to these, there are a few noteworthy &amp;quot;altered&amp;quot; triads that stand out as representations to parts of the overtone series:&lt;br /&gt;
 &lt;br /&gt;
@@ Line 689: / Line 737: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;strong&gt;Ratio&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;Example in C&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;Written name&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;strong&gt;Spoken name&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 697: / Line 751: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;23:27:32&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C Ev Gvv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C.v(vv5)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C dot down, double-down five&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 705: / Line 765: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;7:8:10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C Evv Gv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C.vv(v5)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C dot double-down, down five&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 713: / Line 779: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;9:11:13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C E Gv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C(v5)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C down-five&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 721: / Line 793: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;11:13:17&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C Ev G^&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C.v(^5)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C dot down up-five&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 729: / Line 807: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;13:17:20&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C Fv G^&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C.v4(^5)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C (sus) down-four up-five&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 737: / Line 821: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;11:13:18&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C Ev A&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;A(v5)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;(inversion of 9:11:13)&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 759: / Line 849: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Cents&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Example&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Name*&lt;br /&gt;
@@ Line 767: / Line 859: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(0)-171-343-(514)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C D E F&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Equable diatonic&lt;br /&gt;
@@ Line 775: / Line 869: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(0)-229-400-(514)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C D^ E^ F&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Soft diatonic&lt;br /&gt;
@@ Line 783: / Line 879: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(0)-229-457-(514)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C D^ E^^ F&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Intense diatonic&lt;br /&gt;
@@ Line 791: / Line 889: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(0)-286-457-(514)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C D^^ E^^ F&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Archytas chromatic&lt;br /&gt;
@@ Line 799: / Line 899: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(0)-286-400-(514)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C D^^ E^ F&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Weak chromatic&lt;br /&gt;
@@ Line 807: / Line 909: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(0)-343-457-(514)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C D^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; E^^ F&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Strong enharmonic&lt;br /&gt;
@@ Line 815: / Line 919: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(0)-400-457-(514)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C D^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; E^^ F&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Pythagorean enharmonic&lt;br /&gt;
@@ Line 1,127: / Line 1,233: @@
 &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Books / Literature:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;&lt;strong&gt;Books / Literature:&lt;/strong&gt;&lt;/h1&gt;
   Sword, Ron. &amp;quot;Icosihenaphonic Scales for Guitar&amp;quot;. IAAA Press. 1st ed: July 2009.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:886:&amp;lt;img src=&amp;quot;http://www.ronsword.com/images/ron1.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 188px; width: 254px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:886 --&gt;&lt;!-- ws:start:WikiTextRemoteImageRule:887:&amp;lt;img src=&amp;quot;http://www.swordguitars.com/21tetsm.JPG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 191px; width: 363px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:887 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:944:&amp;lt;img src=&amp;quot;http://www.ronsword.com/images/ron1.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 188px; width: 254px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:944 --&gt;&lt;!-- ws:start:WikiTextRemoteImageRule:945:&amp;lt;img src=&amp;quot;http://www.swordguitars.com/21tetsm.JPG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 191px; width: 363px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:945 --&gt;&lt;br /&gt;
 &lt;strong&gt;&lt;em&gt;21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)&lt;/em&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;br /&gt;