21edo: Difference between revisions

Revision as of 04:20, 26 December 2016

IMPORTED REVISION FROM WIKISPACES

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

This revision was by author TallKite and made on 2016-12-26 04:20:45 UTC.

The original revision id was 602812338.

The revision comment was:

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

Original Wikitext content:

<span style="display: block; text-align: right;">[[21平均律|日本語]]
</span>
=21 equal divisions of the octave= 

Twenty-one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or "equi-heptatonic" scales, or as seven 3-edo ''augmented'' triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other EDO <26.

==21-EDO as a temperament:== 
In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a "third-fourth" (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.

In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit. 21-EDO also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.

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** ||||= [[Ups and Downs Notation|Up/down]]
[[Ups and Downs Notation|Notation]] ||<   ||= **5L3s**
**Octotonic**
**Notation** ||= **D.-R. Interval**
**Types** ||= **Approximate**
**Ratios *1** ||= <span style="display: block; text-align: center;">**Approximate**</span><span style="display: block; text-align: center;">**Ratios *2**</span> ||= <span style="display: block; text-align: center;">**Approximate**</span>**Ratios *3** ||
||= 0 ||= 0 ||> 1 ||= unison ||< C ||= C ||= Unison ||= 1/1 ||= 1/1 ||= 1/1 ||
||= 1 ||= 57.14 ||> ^1
vv2 ||= up unison,
double-down 2nd ||< C^
Dvv ||= C# ||= Subminor 2nd ||= 28/27, 30/29 ||= 35/34, 36/35 ||= 64/63 ||
||= 2 ||= 114.29 ||> ^^1
v2 ||= double-up unison,
down 2nd ||< C^^
Dv ||= Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||= 18/17 ||= 16/15, 25/24 ||
||= 3 ||= 171.43 ||> 2 ||= 2nd ||< D ||= D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 ||
||= 4 ||= 228.57 ||> ^2
vv3 ||= up 2nd,
double-down 3rd ||< D^
Evv ||= D# ||= Supermajor 2nd ||= 8/7 ||= 8/7 ||= 8/7, 10/9, 11/10 ||
||= 5 ||= 285.71 ||> ^^2
v3 ||= double-up 2nd,
down 3rd ||< D^^
Ev ||= Eb ||= Subminor 3rd ||= 27/23, 32/27 ||= 13/11, 20/17 ||= 6/5, 7/6 ||
||= 6 ||= 342.86 ||> 3 ||= 3rd ||< E ||= E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 ||
||= 7 ||= 400 ||> ^3
vv4 ||= up 3rd,
double-down 4th ||< E^
Fvv ||= E#/Fb ||= Major 3rd ||= 29/23 ||= 44/35 ||= 5/4, 9/7, 11/9, 14/11 ||
||= 8 ||= 457.14 ||> ^^3
v4 ||= double-up 3rd,
down 4th ||< E^^
Fv ||= F ||= Third-Fourth ||= 30/23 ||= 13/10, 17/13, 22/17 ||= 13/10 ||
||= 9 ||= 514.29 ||> 4 ||= 4th ||< F ||= F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 ||
||= 10 ||= 571.43 ||> ^4
vv5 ||= up 4th,
double-down 5th ||< F^
Gvv ||= Gb ||= Narrow Tritone ||= 32/23 ||= 18/13 ||= 7/5, 11/8 ||
||= 11 ||= 628.57 ||> ^^4
v5 ||= double-up 4th,
down 5th ||< F^^
Gv ||= G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 ||
||= 12 ||= 685.71 ||> 5 ||= 5th ||< G ||= G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 ||
||= 13 ||= 742.86 ||> ^5
vv6 ||= up 5th,
double-down 6th ||< G^
Avv ||= Hb ||= Fifth-Sixth ||= 23/15 ||= 17/11, 20/13, 26/17 ||= 20/13 ||
||= 14 ||= 800 ||> ^^5
v6 ||= double-up 5th,
down 6th ||< G^^
Av ||= H ||= Minor 6th ||= 46/29 ||= 35/22 ||= 8/5, 11/7, 14/9, 18/11 ||
||= 15 ||= 857.14 ||> 6 ||= 6th ||< A ||= H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 ||
||= 16 ||= 914.29 ||> ^6
vv7 ||= up 6th,
double-down 7th ||< A^
Bvv ||= A ||= Supermajor 6th ||= 27/16, 46/27 ||= 17/10, 22/13 ||= 5/3, 12/7 ||
||= 17 ||= 971.43 ||> ^^6
v7 ||= double-up 6th,
down 7th ||< A^^
Bv ||= A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 ||
||= 18 ||= 1028.57 ||> 7 ||= 7th ||< B ||= Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 ||
||= 19 ||= 1085.71 ||> ^7
vv8 ||= up 7th,
double-down 8ve ||< B^
Cvv ||= B ||= Major 7th ||= 15/8 ||= 17/9 ||= 15/8, 48/25 ||
||= 20 ||= 1142.86 ||> ^^7
v8 ||= double-up 7th,
down 8ve ||< B^^
Cv ||= B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||= 35/18, 68/35 ||= 63/32 ||
||= 21 ||= 1200 ||> 8 ||= 8ve ||< C ||= C ||= Octave ||= 2/1 ||= 2/1 ||= 2/1 ||

*1: based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament
*2: based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament
*3: based on treating 21-EDO as 13-limit laconic temperament

=Chord Names= 

Ups and downs can be used to name 21edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used.

0-6-12 = C E G = C = C or C perfect
0-5-12 = C Ev G = C(v3) = C down-three
0-7-12 = C E^ G = C(^3) = C up-three
0-6-11 = C E Gv = C(v5) = C down-five
0-7-13 = C E^ G^ = C(^3,^5) = C up-three up-five

0-6-12-18 = C E G B = C7 = C seven
0-6-12-17 = C E G Bv = C(v7) = C down-seven
0-5-12-18 = C Ev G B = C7(v3) = C seven down-three
0-5-12-17 = C Ev G Bv = C.v7 = C dot down seven

For a more complete list, see [[xenharmonic/Ups and Downs Notation#Chord%20names%20in%20other%20EDOs|Ups and Downs Notation - Chord names in other EDOs]].

**21-tone scales:**
[[augment6]]
[[augment9]]
[[augment12]]

==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 (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** || **Example in C** || **Written name** || **Spoken name** ||
||= 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 || C Evv Gv || C.vv(v5) || C dot double-down, down five ||
||= 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 || C Ev G^ || C.v(^5) || C dot down up-five ||
||= 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 || C Ev A || A(v5) || (inversion of 9:11:13) ||

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

Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.
For 7-limit harmony (based on a chord of 0-7-12-17 approximating 4:5:6:7), using 1/3-octave period scales (i.e. those related to augmented temperament) yields the most harmonically-efficient scales. The 9-note 3L6s scale (related to Tcherpnin's scale in 12-TET) is an excellent example.

For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.

==Tetrachordal Scales in 21-EDO== 
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 || Example ||= Name* ||< Ups/downs name ||
||= 3, 3, 3 ||= (0)-171-343-(514) || C D E F ||= Equable diatonic ||< C perfect ||
||= 4, 3, 2 ||= (0)-229-400-(514) || C D^ E^ F ||= Soft diatonic ||< C upperfect up-2 ||
||= 4, 4, 1 ||= (0)-229-457-(514) || C D^ E^^ F ||= Intense diatonic ||< C up-2 & 6, double-up-3 & 7 ||
||= 5, 3, 1 ||= (0)-286-457-(514) || C D^^ E^^ F ||= Archytas chromatic ||< C double-up-2, 3, 6 and 7 ||
||= 5, 2, 2 ||= (0)-286-400-(514) || C D^^ E^ F ||= Weak chromatic ||< C double-up 2 & 6, up-3 & 7 ||
||= 6, 2, 1 ||= (0)-343-457-(514) || C D^<span style="font-size: 90%; vertical-align: super;">3</span> E^^ F ||= Strong enharmonic ||< C triple-up 2 & 6, double-up 3 & 7 ||
||= 7, 1, 1 ||= (0)-400-457-(514) || C D^<span style="font-size: 90%; vertical-align: super;">4</span> E^^ F ||= Pythagorean enharmonic ||< C quadruple-up 2 & 6, double-up 3 & 7 ||
*these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!

The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.

==Rank two temperaments== 
[[List of 21edo rank two temperaments by badness]]
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\21 || [[Escapade family#Escapade|Escapade]] ||
|| 1 || 2\21 || [[Gamelismic clan#Miracle|Miracle]] ||
|| 1 || 4\21 || [[Slendric]]/[[Gamelismic clan#Gorgo|Gorgo]]/[[Gamelismic clan#Gidorah|Gidorah]] ||
|| 1 || 5\21 || [[Mint temperaments#Subklei|Subklei]] ||
|| 1 || 8\21 || [[Chromatic pairs#Tridec|Tridec]] ||
|| 1 || 10\21 || [[Marvel temperaments#Triton|Triton]] ||
|| 3 || 1\21 ||   ||
|| 3 || 2\21 || [[Augmented family|Augmented]]/[[August]] ||
|| 3 || 3\21 || [[Oodako]] ||
|| 7 || 1\21 || [[Apotome family|Whitewood]] ||


==13-limit Commas== 
21 EDO tempers out the following 13-limit commas. (Note: This assumes the val < 21 33 49 59 73 78 |.)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||
||= 2187/2048 ||< | -11 7 > ||> 113.69 ||= Apotome ||=   ||
||= 128/125 ||< | 7 0 -3 > ||> 41.06 ||= Diesis ||= Augmented Comma ||
||=   ||< | -25 7 6 > ||> 31.57 ||= Ampersand's Comma ||   ||
||=   ||< | 32 -7 -9 > ||> 9.49 ||= Escapade Comma ||   ||
||= 1029/1000 ||< | -3 1 -3 3 > ||> 49.49 ||= Keega ||=   ||
||= 36/35 ||< | 2 2 -1 -1 > ||> 48.77 ||= Septimal Quarter Tone ||   ||
||=   ||< | -10 7 8 -7 > ||> 22.41 ||= Blackjackisma ||=   ||
||= 1029/1024 ||< | -10 1 0 3 > ||> 8.43 ||= Gamelisma ||=   ||
||= 225/224 ||< | -5 2 2 -1 > ||> 7.71 ||= Septimal Kleisma ||= Marvel Comma ||
||= 16875/16807 ||< | 0 3 4 -5 > ||> 6.99 ||= Mirkwai ||=   ||
||= 2401/2400 ||< | -5 -1 -2 4 > ||> 0.72 ||= Breedsma ||=   ||
||=   ||< | 47 -7 -7 -7 > ||> 0.34 ||= Akjaysma ||= 5\7 Octave Comma ||
||= 99/98 ||< | -1 2 0 -2 1 > ||> 17.58 ||= Mothwellsma ||=   ||
||= 176/175 ||< | 4 0 -2 -1 1 > ||> 9.86 ||= Valinorsma ||=   ||
||= 4000/3993 ||< | 5 -1 3 0 -3 > ||> 3.03 ||= Wizardharry ||=   ||



=**Books / Literature:**= 
Sword, Ron. "Icosihenaphonic Scales for Guitar". IAAA Press. 1st ed: July 2009.
[[image:http://www.ronsword.com/images/ron1.jpg width="254" height="188"]][[image:http://www.swordguitars.com/21tetsm.JPG width="363" height="191"]]
**//21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)//**

=**Compositions/Listening:**= 
[[@http://soonlabel.com/xenharmonic/archives/2494|21-edo Trio for Organ, by Claudi Meneghin]]
[[@http://soonlabel.com/xenharmonic/archives/2336|21-penny jingle, by Claudi Meneghin]]
[[@http://www.ronsword.com/sounds/21_improv.mp3|Short Clip of 21-edo Acoustic]] by [[Ron Sword]]
[[@http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3|Open tuning Drone Improvisation in 21-edo]] by Ron Sword
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=933715|Anomalous Readings]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3|play]] by [[Andrew Heathwaite]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/15%20-%2015.%2021%20octave.mp3|Comets Over Flatland 15]] by [[Randy Winchester]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/18%20-%2018.%2021%20octave.mp3|Comets Over Flatland 18]] by [[Randy Winchester]]
[[@http://www.reverbnation.com/ffffiale/song/17858773-lesatonale-ubriaco|L'esatonale ubriaco (the drunk hexatonal)]], ALIENAMENTE by [[xenharmonic/Fabrizio Fiale|Fabrizio Fulvio Fausto Fiale]]

Original HTML content:

<html><head><title>21edo</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="/21%E5%B9%B3%E5%9D%87%E5%BE%8B">日本語</a><br />
</span><br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x21 equal divisions of the octave"></a><!-- ws:end:WikiTextHeadingRule:0 -->21 equal divisions of the octave</h1>
 <br />
Twenty-one equal divisions of the octave provides the sonic fingerprint of the augmented and 7-edo family, while also giving a some higher harmony possibilities and fun intervals like the apotome. The system can be treated as three intertwining 7-edo or &quot;equi-heptatonic&quot; scales, or as seven 3-edo ''augmented'' triads. The 7/4 at 968.826 cents is only off in 21-tone by 2.6 cents, which is better than any other EDO &lt;26.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x21 equal divisions of the octave-21-EDO as a temperament:"></a><!-- ws:end:WikiTextHeadingRule:2 -->21-EDO as a temperament:</h2>
 In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a &quot;third-fourth&quot; (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.<br />
<br />
In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit. 21-EDO also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.<br />
<br />
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 <a class="wiki_link" href="/Laconic%20Family">laconic</a> 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 &quot;interesting&quot; 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.<br />
<br />


<table class="wiki_table">
    <tr>
        <td style="text-align: center;"><strong>Degree</strong><br />
</td>
        <td style="text-align: center;"><strong>Cents</strong><br />
</td>
        <td colspan="2" style="text-align: center;"><a class="wiki_link" href="/Ups%20and%20Downs%20Notation">Up/down</a><br />
<a class="wiki_link" href="/Ups%20and%20Downs%20Notation">Notation</a><br />
</td>
        <td style="text-align: left;"><br />
</td>
        <td style="text-align: center;"><strong>5L3s</strong><br />
<strong>Octotonic</strong><br />
<strong>Notation</strong><br />
</td>
        <td style="text-align: center;"><strong>D.-R. Interval</strong><br />
<strong>Types</strong><br />
</td>
        <td style="text-align: center;"><strong>Approximate</strong><br />
<strong>Ratios *1</strong><br />
</td>
        <td style="text-align: center;"><span style="display: block; text-align: center;"><strong>Approximate</strong></span><span style="display: block; text-align: center;"><strong>Ratios *2</strong></span><br />
</td>
        <td style="text-align: center;"><span style="display: block; text-align: center;"><strong>Approximate</strong></span><strong>Ratios *3</strong><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: right;">1<br />
</td>
        <td style="text-align: center;">unison<br />
</td>
        <td style="text-align: left;">C<br />
</td>
        <td style="text-align: center;">C<br />
</td>
        <td style="text-align: center;">Unison<br />
</td>
        <td style="text-align: center;">1/1<br />
</td>
        <td style="text-align: center;">1/1<br />
</td>
        <td style="text-align: center;">1/1<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: center;">57.14<br />
</td>
        <td style="text-align: right;">^1<br />
vv2<br />
</td>
        <td style="text-align: center;">up unison,<br />
double-down 2nd<br />
</td>
        <td style="text-align: left;">C^<br />
Dvv<br />
</td>
        <td style="text-align: center;">C#<br />
</td>
        <td style="text-align: center;">Subminor 2nd<br />
</td>
        <td style="text-align: center;">28/27, 30/29<br />
</td>
        <td style="text-align: center;">35/34, 36/35<br />
</td>
        <td style="text-align: center;">64/63<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: center;">114.29<br />
</td>
        <td style="text-align: right;">^^1<br />
v2<br />
</td>
        <td style="text-align: center;">double-up unison,<br />
down 2nd<br />
</td>
        <td style="text-align: left;">C^^<br />
Dv<br />
</td>
        <td style="text-align: center;">Db<br />
</td>
        <td style="text-align: center;">Minor 2nd<br />
</td>
        <td style="text-align: center;">16/15, 15/14, 29/27<br />
</td>
        <td style="text-align: center;">18/17<br />
</td>
        <td style="text-align: center;">16/15, 25/24<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: center;">171.43<br />
</td>
        <td style="text-align: right;">2<br />
</td>
        <td style="text-align: center;">2nd<br />
</td>
        <td style="text-align: left;">D<br />
</td>
        <td style="text-align: center;">D<br />
</td>
        <td style="text-align: center;">Submajor 2nd<br />
</td>
        <td style="text-align: center;">10/9, 32/29<br />
</td>
        <td style="text-align: center;">10/9,11/10<br />
</td>
        <td style="text-align: center;">9/8<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: center;">228.57<br />
</td>
        <td style="text-align: right;">^2<br />
vv3<br />
</td>
        <td style="text-align: center;">up 2nd,<br />
double-down 3rd<br />
</td>
        <td style="text-align: left;">D^<br />
Evv<br />
</td>
        <td style="text-align: center;">D#<br />
</td>
        <td style="text-align: center;">Supermajor 2nd<br />
</td>
        <td style="text-align: center;">8/7<br />
</td>
        <td style="text-align: center;">8/7<br />
</td>
        <td style="text-align: center;">8/7, 10/9, 11/10<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: center;">285.71<br />
</td>
        <td style="text-align: right;">^^2<br />
v3<br />
</td>
        <td style="text-align: center;">double-up 2nd,<br />
down 3rd<br />
</td>
        <td style="text-align: left;">D^^<br />
Ev<br />
</td>
        <td style="text-align: center;">Eb<br />
</td>
        <td style="text-align: center;">Subminor 3rd<br />
</td>
        <td style="text-align: center;">27/23, 32/27<br />
</td>
        <td style="text-align: center;">13/11, 20/17<br />
</td>
        <td style="text-align: center;">6/5, 7/6<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6<br />
</td>
        <td style="text-align: center;">342.86<br />
</td>
        <td style="text-align: right;">3<br />
</td>
        <td style="text-align: center;">3rd<br />
</td>
        <td style="text-align: left;">E<br />
</td>
        <td style="text-align: center;">E<br />
</td>
        <td style="text-align: center;">Neutral 3rd<br />
</td>
        <td style="text-align: center;">28/23<br />
</td>
        <td style="text-align: center;">11/9<br />
</td>
        <td style="text-align: center;">16/13<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">7<br />
</td>
        <td style="text-align: center;">400<br />
</td>
        <td style="text-align: right;">^3<br />
vv4<br />
</td>
        <td style="text-align: center;">up 3rd,<br />
double-down 4th<br />
</td>
        <td style="text-align: left;">E^<br />
Fvv<br />
</td>
        <td style="text-align: center;">E#/Fb<br />
</td>
        <td style="text-align: center;">Major 3rd<br />
</td>
        <td style="text-align: center;">29/23<br />
</td>
        <td style="text-align: center;">44/35<br />
</td>
        <td style="text-align: center;">5/4, 9/7, 11/9, 14/11<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">8<br />
</td>
        <td style="text-align: center;">457.14<br />
</td>
        <td style="text-align: right;">^^3<br />
v4<br />
</td>
        <td style="text-align: center;">double-up 3rd,<br />
down 4th<br />
</td>
        <td style="text-align: left;">E^^<br />
Fv<br />
</td>
        <td style="text-align: center;">F<br />
</td>
        <td style="text-align: center;">Third-Fourth<br />
</td>
        <td style="text-align: center;">30/23<br />
</td>
        <td style="text-align: center;">13/10, 17/13, 22/17<br />
</td>
        <td style="text-align: center;">13/10<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9<br />
</td>
        <td style="text-align: center;">514.29<br />
</td>
        <td style="text-align: right;">4<br />
</td>
        <td style="text-align: center;">4th<br />
</td>
        <td style="text-align: left;">F<br />
</td>
        <td style="text-align: center;">F#<br />
</td>
        <td style="text-align: center;">Acute 4th<br />
</td>
        <td style="text-align: center;">161/120, 256/189<br />
</td>
        <td style="text-align: center;">35/26<br />
</td>
        <td style="text-align: center;">4/3, 18/13<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">10<br />
</td>
        <td style="text-align: center;">571.43<br />
</td>
        <td style="text-align: right;">^4<br />
vv5<br />
</td>
        <td style="text-align: center;">up 4th,<br />
double-down 5th<br />
</td>
        <td style="text-align: left;">F^<br />
Gvv<br />
</td>
        <td style="text-align: center;">Gb<br />
</td>
        <td style="text-align: center;">Narrow Tritone<br />
</td>
        <td style="text-align: center;">32/23<br />
</td>
        <td style="text-align: center;">18/13<br />
</td>
        <td style="text-align: center;">7/5, 11/8<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">11<br />
</td>
        <td style="text-align: center;">628.57<br />
</td>
        <td style="text-align: right;">^^4<br />
v5<br />
</td>
        <td style="text-align: center;">double-up 4th,<br />
down 5th<br />
</td>
        <td style="text-align: left;">F^^<br />
Gv<br />
</td>
        <td style="text-align: center;">G<br />
</td>
        <td style="text-align: center;">Wide Tritone<br />
</td>
        <td style="text-align: center;">23/16<br />
</td>
        <td style="text-align: center;">13/9<br />
</td>
        <td style="text-align: center;">10/7, 16/11<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">12<br />
</td>
        <td style="text-align: center;">685.71<br />
</td>
        <td style="text-align: right;">5<br />
</td>
        <td style="text-align: center;">5th<br />
</td>
        <td style="text-align: left;">G<br />
</td>
        <td style="text-align: center;">G#<br />
</td>
        <td style="text-align: center;">Grave 5th<br />
</td>
        <td style="text-align: center;">189/128, 240/161<br />
</td>
        <td style="text-align: center;">52/35<br />
</td>
        <td style="text-align: center;">3/2, 13/9<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">13<br />
</td>
        <td style="text-align: center;">742.86<br />
</td>
        <td style="text-align: right;">^5<br />
vv6<br />
</td>
        <td style="text-align: center;">up 5th,<br />
double-down 6th<br />
</td>
        <td style="text-align: left;">G^<br />
Avv<br />
</td>
        <td style="text-align: center;">Hb<br />
</td>
        <td style="text-align: center;">Fifth-Sixth<br />
</td>
        <td style="text-align: center;">23/15<br />
</td>
        <td style="text-align: center;">17/11, 20/13, 26/17<br />
</td>
        <td style="text-align: center;">20/13<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">14<br />
</td>
        <td style="text-align: center;">800<br />
</td>
        <td style="text-align: right;">^^5<br />
v6<br />
</td>
        <td style="text-align: center;">double-up 5th,<br />
down 6th<br />
</td>
        <td style="text-align: left;">G^^<br />
Av<br />
</td>
        <td style="text-align: center;">H<br />
</td>
        <td style="text-align: center;">Minor 6th<br />
</td>
        <td style="text-align: center;">46/29<br />
</td>
        <td style="text-align: center;">35/22<br />
</td>
        <td style="text-align: center;">8/5, 11/7, 14/9, 18/11<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">15<br />
</td>
        <td style="text-align: center;">857.14<br />
</td>
        <td style="text-align: right;">6<br />
</td>
        <td style="text-align: center;">6th<br />
</td>
        <td style="text-align: left;">A<br />
</td>
        <td style="text-align: center;">H#/Ab<br />
</td>
        <td style="text-align: center;">Neutral 6th<br />
</td>
        <td style="text-align: center;">23/14<br />
</td>
        <td style="text-align: center;">18/11<br />
</td>
        <td style="text-align: center;">13/8<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16<br />
</td>
        <td style="text-align: center;">914.29<br />
</td>
        <td style="text-align: right;">^6<br />
vv7<br />
</td>
        <td style="text-align: center;">up 6th,<br />
double-down 7th<br />
</td>
        <td style="text-align: left;">A^<br />
Bvv<br />
</td>
        <td style="text-align: center;">A<br />
</td>
        <td style="text-align: center;">Supermajor 6th<br />
</td>
        <td style="text-align: center;">27/16, 46/27<br />
</td>
        <td style="text-align: center;">17/10, 22/13<br />
</td>
        <td style="text-align: center;">5/3, 12/7<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">17<br />
</td>
        <td style="text-align: center;">971.43<br />
</td>
        <td style="text-align: right;">^^6<br />
v7<br />
</td>
        <td style="text-align: center;">double-up 6th,<br />
down 7th<br />
</td>
        <td style="text-align: left;">A^^<br />
Bv<br />
</td>
        <td style="text-align: center;">A#<br />
</td>
        <td style="text-align: center;">Subminor 7th<br />
</td>
        <td style="text-align: center;">7/4<br />
</td>
        <td style="text-align: center;">7/4<br />
</td>
        <td style="text-align: center;">7/4, 9/5, 20/11<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">18<br />
</td>
        <td style="text-align: center;">1028.57<br />
</td>
        <td style="text-align: right;">7<br />
</td>
        <td style="text-align: center;">7th<br />
</td>
        <td style="text-align: left;">B<br />
</td>
        <td style="text-align: center;">Bb<br />
</td>
        <td style="text-align: center;">Supraminor 7th<br />
</td>
        <td style="text-align: center;">29/16, 9/5<br />
</td>
        <td style="text-align: center;">9/5, 20/11<br />
</td>
        <td style="text-align: center;">16/9<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">19<br />
</td>
        <td style="text-align: center;">1085.71<br />
</td>
        <td style="text-align: right;">^7<br />
vv8<br />
</td>
        <td style="text-align: center;">up 7th,<br />
double-down 8ve<br />
</td>
        <td style="text-align: left;">B^<br />
Cvv<br />
</td>
        <td style="text-align: center;">B<br />
</td>
        <td style="text-align: center;">Major 7th<br />
</td>
        <td style="text-align: center;">15/8<br />
</td>
        <td style="text-align: center;">17/9<br />
</td>
        <td style="text-align: center;">15/8, 48/25<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">20<br />
</td>
        <td style="text-align: center;">1142.86<br />
</td>
        <td style="text-align: right;">^^7<br />
v8<br />
</td>
        <td style="text-align: center;">double-up 7th,<br />
down 8ve<br />
</td>
        <td style="text-align: left;">B^^<br />
Cv<br />
</td>
        <td style="text-align: center;">B#/Cb<br />
</td>
        <td style="text-align: center;">Supermajor 7th<br />
</td>
        <td style="text-align: center;">27/14, 29/15<br />
</td>
        <td style="text-align: center;">35/18, 68/35<br />
</td>
        <td style="text-align: center;">63/32<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">21<br />
</td>
        <td style="text-align: center;">1200<br />
</td>
        <td style="text-align: right;">8<br />
</td>
        <td style="text-align: center;">8ve<br />
</td>
        <td style="text-align: left;">C<br />
</td>
        <td style="text-align: center;">C<br />
</td>
        <td style="text-align: center;">Octave<br />
</td>
        <td style="text-align: center;">2/1<br />
</td>
        <td style="text-align: center;">2/1<br />
</td>
        <td style="text-align: center;">2/1<br />
</td>
    </tr>
</table>

<br />
*1: based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament<br />
*2: based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament<br />
*3: based on treating 21-EDO as 13-limit laconic temperament<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Chord Names"></a><!-- ws:end:WikiTextHeadingRule:4 -->Chord Names</h1>
 <br />
Ups and downs can be used to name 21edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used.<br />
<br />
0-6-12 = C E G = C = C or C perfect<br />
0-5-12 = C Ev G = C(v3) = C down-three<br />
0-7-12 = C E^ G = C(^3) = C up-three<br />
0-6-11 = C E Gv = C(v5) = C down-five<br />
0-7-13 = C E^ G^ = C(^3,^5) = C up-three up-five<br />
<br />
0-6-12-18 = C E G B = C7 = C seven<br />
0-6-12-17 = C E G Bv = C(v7) = C down-seven<br />
0-5-12-18 = C Ev G B = C7(v3) = C seven down-three<br />
0-5-12-17 = C Ev G Bv = C.v7 = C dot down seven<br />
<br />
For a more complete list, see <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Chord%20names%20in%20other%20EDOs">Ups and Downs Notation - Chord names in other EDOs</a>.<br />
<br />
<strong>21-tone scales:</strong><br />
<a class="wiki_link" href="/augment6">augment6</a><br />
<a class="wiki_link" href="/augment9">augment9</a><br />
<a class="wiki_link" href="/augment12">augment12</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Chord Names-Triadic Harmony in 21-EDO:"></a><!-- ws:end:WikiTextHeadingRule:6 -->Triadic Harmony in 21-EDO:</h2>
 <br />
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:<br />
<br />


<table class="wiki_table">
    <tr>
        <td style="text-align: center;"><strong>Steps</strong><br />
</td>
        <td style="text-align: center;"><strong>Cents</strong><br />
</td>
        <td style="text-align: center;"><strong>Ratio</strong><br />
</td>
        <td><strong>Example in C</strong><br />
</td>
        <td><strong>Written name</strong><br />
</td>
        <td><strong>Spoken name</strong><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">0-5-10<br />
</td>
        <td style="text-align: center;">0-286-571<br />
</td>
        <td style="text-align: center;">23:27:32<br />
</td>
        <td>C Ev Gvv<br />
</td>
        <td>C.v(vv5)<br />
</td>
        <td>C dot down, double-down five<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">0-4-11<br />
</td>
        <td style="text-align: center;">0-229-629<br />
</td>
        <td style="text-align: center;">7:8:10<br />
</td>
        <td>C Evv Gv<br />
</td>
        <td>C.vv(v5)<br />
</td>
        <td>C dot double-down, down five<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">0-6-11<br />
</td>
        <td style="text-align: center;">0-343-629<br />
</td>
        <td style="text-align: center;">9:11:13<br />
</td>
        <td>C E Gv<br />
</td>
        <td>C(v5)<br />
</td>
        <td>C down-five<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">0-5-13<br />
</td>
        <td style="text-align: center;">0-286-743<br />
</td>
        <td style="text-align: center;">11:13:17<br />
</td>
        <td>C Ev G^<br />
</td>
        <td>C.v(^5)<br />
</td>
        <td>C dot down up-five<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">0-8-13<br />
</td>
        <td style="text-align: center;">0-457-743<br />
</td>
        <td style="text-align: center;">13:17:20<br />
</td>
        <td>C Fv G^<br />
</td>
        <td>C.v4(^5)<br />
</td>
        <td>C (sus) down-four up-five<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">0-5-15<br />
</td>
        <td style="text-align: center;">0-286-857<br />
</td>
        <td style="text-align: center;">11:13:18<br />
</td>
        <td>C Ev A<br />
</td>
        <td>A(v5)<br />
</td>
        <td>(inversion of 9:11:13)<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Chord Names-Moment-of-Symmetry Scales in 21-EDO:"></a><!-- ws:end:WikiTextHeadingRule:8 -->Moment-of-Symmetry Scales in 21-EDO:</h2>
 <br />
Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.<br />
For 7-limit harmony (based on a chord of 0-7-12-17 approximating 4:5:6:7), using 1/3-octave period scales (i.e. those related to augmented temperament) yields the most harmonically-efficient scales. The 9-note 3L6s scale (related to Tcherpnin's scale in 12-TET) is an excellent example.<br />
<br />
For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Chord Names-Tetrachordal Scales in 21-EDO"></a><!-- ws:end:WikiTextHeadingRule:10 -->Tetrachordal Scales in 21-EDO</h2>
 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:<br />
<br />


<table class="wiki_table">
    <tr>
        <td style="text-align: center;">Step Pattern<br />
</td>
        <td style="text-align: center;">Cents<br />
</td>
        <td>Example<br />
</td>
        <td style="text-align: center;">Name*<br />
</td>
        <td style="text-align: left;">Ups/downs name<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3, 3, 3<br />
</td>
        <td style="text-align: center;">(0)-171-343-(514)<br />
</td>
        <td>C D E F<br />
</td>
        <td style="text-align: center;">Equable diatonic<br />
</td>
        <td style="text-align: left;">C perfect<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4, 3, 2<br />
</td>
        <td style="text-align: center;">(0)-229-400-(514)<br />
</td>
        <td>C D^ E^ F<br />
</td>
        <td style="text-align: center;">Soft diatonic<br />
</td>
        <td style="text-align: left;">C upperfect up-2<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4, 4, 1<br />
</td>
        <td style="text-align: center;">(0)-229-457-(514)<br />
</td>
        <td>C D^ E^^ F<br />
</td>
        <td style="text-align: center;">Intense diatonic<br />
</td>
        <td style="text-align: left;">C up-2 &amp; 6, double-up-3 &amp; 7<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">5, 3, 1<br />
</td>
        <td style="text-align: center;">(0)-286-457-(514)<br />
</td>
        <td>C D^^ E^^ F<br />
</td>
        <td style="text-align: center;">Archytas chromatic<br />
</td>
        <td style="text-align: left;">C double-up-2, 3, 6 and 7<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">5, 2, 2<br />
</td>
        <td style="text-align: center;">(0)-286-400-(514)<br />
</td>
        <td>C D^^ E^ F<br />
</td>
        <td style="text-align: center;">Weak chromatic<br />
</td>
        <td style="text-align: left;">C double-up 2 &amp; 6, up-3 &amp; 7<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6, 2, 1<br />
</td>
        <td style="text-align: center;">(0)-343-457-(514)<br />
</td>
        <td>C D^<span style="font-size: 90%; vertical-align: super;">3</span> E^^ F<br />
</td>
        <td style="text-align: center;">Strong enharmonic<br />
</td>
        <td style="text-align: left;">C triple-up 2 &amp; 6, double-up 3 &amp; 7<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">7, 1, 1<br />
</td>
        <td style="text-align: center;">(0)-400-457-(514)<br />
</td>
        <td>C D^<span style="font-size: 90%; vertical-align: super;">4</span> E^^ F<br />
</td>
        <td style="text-align: center;">Pythagorean enharmonic<br />
</td>
        <td style="text-align: left;">C quadruple-up 2 &amp; 6, double-up 3 &amp; 7<br />
</td>
    </tr>
</table>

*these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!<br />
<br />
The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Chord Names-Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:12 -->Rank two temperaments</h2>
 <a class="wiki_link" href="/List%20of%2021edo%20rank%20two%20temperaments%20by%20badness">List of 21edo rank two temperaments by badness</a><br />


<table class="wiki_table">
    <tr>
        <th>Periods<br />
per octave<br />
</th>
        <th>Generator<br />
</th>
        <th>Temperaments<br />
</th>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>1\21<br />
</td>
        <td><a class="wiki_link" href="/Escapade%20family#Escapade">Escapade</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>2\21<br />
</td>
        <td><a class="wiki_link" href="/Gamelismic%20clan#Miracle">Miracle</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>4\21<br />
</td>
        <td><a class="wiki_link" href="/Slendric">Slendric</a>/<a class="wiki_link" href="/Gamelismic%20clan#Gorgo">Gorgo</a>/<a class="wiki_link" href="/Gamelismic%20clan#Gidorah">Gidorah</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>5\21<br />
</td>
        <td><a class="wiki_link" href="/Mint%20temperaments#Subklei">Subklei</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>8\21<br />
</td>
        <td><a class="wiki_link" href="/Chromatic%20pairs#Tridec">Tridec</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>10\21<br />
</td>
        <td><a class="wiki_link" href="/Marvel%20temperaments#Triton">Triton</a><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>1\21<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>2\21<br />
</td>
        <td><a class="wiki_link" href="/Augmented%20family">Augmented</a>/<a class="wiki_link" href="/August">August</a><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>3\21<br />
</td>
        <td><a class="wiki_link" href="/Oodako">Oodako</a><br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>1\21<br />
</td>
        <td><a class="wiki_link" href="/Apotome%20family">Whitewood</a><br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Chord Names-13-limit Commas"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit Commas</h2>
 21 EDO tempers out the following 13-limit commas. (Note: This assumes the val &lt; 21 33 49 59 73 78 |.)<br />


<table class="wiki_table">
    <tr>
        <th>Comma<br />
</th>
        <th>Monzo<br />
</th>
        <th>Value (Cents)<br />
</th>
        <th>Name 1<br />
</th>
        <th>Name 2<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">2187/2048<br />
</td>
        <td style="text-align: left;">| -11 7 &gt;<br />
</td>
        <td style="text-align: right;">113.69<br />
</td>
        <td style="text-align: center;">Apotome<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">128/125<br />
</td>
        <td style="text-align: left;">| 7 0 -3 &gt;<br />
</td>
        <td style="text-align: right;">41.06<br />
</td>
        <td style="text-align: center;">Diesis<br />
</td>
        <td style="text-align: center;">Augmented Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: left;">| -25 7 6 &gt;<br />
</td>
        <td style="text-align: right;">31.57<br />
</td>
        <td style="text-align: center;">Ampersand's Comma<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: left;">| 32 -7 -9 &gt;<br />
</td>
        <td style="text-align: right;">9.49<br />
</td>
        <td style="text-align: center;">Escapade Comma<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1029/1000<br />
</td>
        <td style="text-align: left;">| -3 1 -3 3 &gt;<br />
</td>
        <td style="text-align: right;">49.49<br />
</td>
        <td style="text-align: center;">Keega<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">36/35<br />
</td>
        <td style="text-align: left;">| 2 2 -1 -1 &gt;<br />
</td>
        <td style="text-align: right;">48.77<br />
</td>
        <td style="text-align: center;">Septimal Quarter Tone<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: left;">| -10 7 8 -7 &gt;<br />
</td>
        <td style="text-align: right;">22.41<br />
</td>
        <td style="text-align: center;">Blackjackisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1029/1024<br />
</td>
        <td style="text-align: left;">| -10 1 0 3 &gt;<br />
</td>
        <td style="text-align: right;">8.43<br />
</td>
        <td style="text-align: center;">Gamelisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">225/224<br />
</td>
        <td style="text-align: left;">| -5 2 2 -1 &gt;<br />
</td>
        <td style="text-align: right;">7.71<br />
</td>
        <td style="text-align: center;">Septimal Kleisma<br />
</td>
        <td style="text-align: center;">Marvel Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16875/16807<br />
</td>
        <td style="text-align: left;">| 0 3 4 -5 &gt;<br />
</td>
        <td style="text-align: right;">6.99<br />
</td>
        <td style="text-align: center;">Mirkwai<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">2401/2400<br />
</td>
        <td style="text-align: left;">| -5 -1 -2 4 &gt;<br />
</td>
        <td style="text-align: right;">0.72<br />
</td>
        <td style="text-align: center;">Breedsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: left;">| 47 -7 -7 -7 &gt;<br />
</td>
        <td style="text-align: right;">0.34<br />
</td>
        <td style="text-align: center;">Akjaysma<br />
</td>
        <td style="text-align: center;">5\7 Octave Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">99/98<br />
</td>
        <td style="text-align: left;">| -1 2 0 -2 1 &gt;<br />
</td>
        <td style="text-align: right;">17.58<br />
</td>
        <td style="text-align: center;">Mothwellsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">176/175<br />
</td>
        <td style="text-align: left;">| 4 0 -2 -1 1 &gt;<br />
</td>
        <td style="text-align: right;">9.86<br />
</td>
        <td style="text-align: center;">Valinorsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4000/3993<br />
</td>
        <td style="text-align: left;">| 5 -1 3 0 -3 &gt;<br />
</td>
        <td style="text-align: right;">3.03<br />
</td>
        <td style="text-align: center;">Wizardharry<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Books / Literature:"></a><!-- ws:end:WikiTextHeadingRule:16 --><strong>Books / Literature:</strong></h1>
 Sword, Ron. &quot;Icosihenaphonic Scales for Guitar&quot;. IAAA Press. 1st ed: July 2009.<br />
<!-- ws:start:WikiTextRemoteImageRule:1008:&lt;img src=&quot;http://www.ronsword.com/images/ron1.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 188px; width: 254px;&quot; /&gt; --><img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /><!-- ws:end:WikiTextRemoteImageRule:1008 --><!-- ws:start:WikiTextRemoteImageRule:1009:&lt;img src=&quot;http://www.swordguitars.com/21tetsm.JPG&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 191px; width: 363px;&quot; /&gt; --><img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" /><!-- ws:end:WikiTextRemoteImageRule:1009 --><br />
<strong><em>21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)</em></strong><br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc9"><a name="Compositions/Listening:"></a><!-- ws:end:WikiTextHeadingRule:18 --><strong>Compositions/Listening:</strong></h1>
 <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/2494" rel="nofollow" target="_blank">21-edo Trio for Organ, by Claudi Meneghin</a><br />
<a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/2336" rel="nofollow" target="_blank">21-penny jingle, by Claudi Meneghin</a><br />
<a class="wiki_link_ext" href="http://www.ronsword.com/sounds/21_improv.mp3" rel="nofollow" target="_blank">Short Clip of 21-edo Acoustic</a> by <a class="wiki_link" href="/Ron%20Sword">Ron Sword</a><br />
<a class="wiki_link_ext" href="http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3" rel="nofollow" target="_blank">Open tuning Drone Improvisation in 21-edo</a> by Ron Sword<br />
<a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933715" rel="nofollow">Anomalous Readings</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/15%20-%2015.%2021%20octave.mp3" rel="nofollow">Comets Over Flatland 15</a> by <a class="wiki_link" href="/Randy%20Winchester">Randy Winchester</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/18%20-%2018.%2021%20octave.mp3" rel="nofollow">Comets Over Flatland 18</a> by <a class="wiki_link" href="/Randy%20Winchester">Randy Winchester</a><br />
<a class="wiki_link_ext" href="http://www.reverbnation.com/ffffiale/song/17858773-lesatonale-ubriaco" rel="nofollow" target="_blank">L'esatonale ubriaco (the drunk hexatonal)</a>, ALIENAMENTE by <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Fabrizio%20Fiale">Fabrizio Fulvio Fausto Fiale</a></body></html>

@@ 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>2016-12-22 04:46:43 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-26 04:20:45 UTC</tt>.<br>
-: The original revision id was <tt>602739102</tt>.<br>
+: The original revision id was <tt>602812338</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 25: / Line 25: @@
 **Types** ||= **Approximate**
 **Ratios *1** ||= &lt;span style="display: block; text-align: center;"&gt;**Approximate**&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;**Ratios *2**&lt;/span&gt; ||= &lt;span style="display: block; text-align: center;"&gt;**Approximate**&lt;/span&gt;**Ratios *3** ||
-||= 0 ||= 0 ||&gt; 1 ||= _____unison_____ ||&lt; C ||= C ||= Unison ||= 1/1 ||= 1/1 ||= 1/1 ||
+||= 0 ||= 0 ||&gt; 1 ||= unison ||&lt; C ||= C ||= Unison ||= 1/1 ||= 1/1 ||= 1/1 ||
 ||= 1 ||= 57.14 ||&gt; ^1
 vv2 ||= up unison,
@@ Line 34: / Line 34: @@
 down 2nd ||&lt; C^^
 Dv ||= Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||= 18/17 ||= 16/15, 25/24 ||
-||= 3 ||= 171.43 ||&gt; 2 ||= perfect 2nd ||&lt; D ||= D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 ||
+||= 3 ||= 171.43 ||&gt; 2 ||= 2nd ||&lt; D ||= D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 ||
 ||= 4 ||= 228.57 ||&gt; ^2
 vv3 ||= up 2nd,
@@ Line 43: / Line 43: @@
 down 3rd ||&lt; D^^
 Ev ||= Eb ||= Subminor 3rd ||= 27/23, 32/27 ||= 13/11, 20/17 ||= 6/5, 7/6 ||
-||= 6 ||= 342.86 ||&gt; 3 ||= perfect 3rd ||&lt; E ||= E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 ||
+||= 6 ||= 342.86 ||&gt; 3 ||= 3rd ||&lt; E ||= E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 ||
 ||= 7 ||= 400 ||&gt; ^3
 vv4 ||= up 3rd,
@@ Line 52: / Line 52: @@
 down 4th ||&lt; E^^
 Fv ||= F ||= Third-Fourth ||= 30/23 ||= 13/10, 17/13, 22/17 ||= 13/10 ||
-||= 9 ||= 514.29 ||&gt; 4 ||= perfect 4th ||&lt; F ||= F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 ||
+||= 9 ||= 514.29 ||&gt; 4 ||= 4th ||&lt; F ||= F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 ||
 ||= 10 ||= 571.43 ||&gt; ^4
 vv5 ||= up 4th,
@@ Line 61: / Line 61: @@
 down 5th ||&lt; F^^
 Gv ||= G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 ||
-||= 12 ||= 685.71 ||&gt; 5 ||= perfect 5th ||&lt; G ||= G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 ||
+||= 12 ||= 685.71 ||&gt; 5 ||= 5th ||&lt; G ||= G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 ||
 ||= 13 ||= 742.86 ||&gt; ^5
 vv6 ||= up 5th,
@@ Line 70: / Line 70: @@
 down 6th ||&lt; G^^
 Av ||= H ||= Minor 6th ||= 46/29 ||= 35/22 ||= 8/5, 11/7, 14/9, 18/11 ||
-||= 15 ||= 857.14 ||&gt; 6 ||= perfect 6th ||&lt; A ||= H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 ||
+||= 15 ||= 857.14 ||&gt; 6 ||= 6th ||&lt; A ||= H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 ||
 ||= 16 ||= 914.29 ||&gt; ^6
 vv7 ||= up 6th,
@@ Line 79: / Line 79: @@
 down 7th ||&lt; A^^
 Bv ||= A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 ||
-||= 18 ||= 1028.57 ||&gt; 7 ||= perfect 7th ||&lt; B ||= Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 ||
+||= 18 ||= 1028.57 ||&gt; 7 ||= 7th ||&lt; B ||= Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 ||
 ||= 19 ||= 1085.71 ||&gt; ^7
 vv8 ||= up 7th,
@@ Line 93: / Line 93: @@
 *2: based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament
 *3: based on treating 21-EDO as 13-limit laconic temperament
+=Chord Names=
+Ups and downs can be used to name 21edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used.
+-6-12 = C E G = C = C or C perfect
+-5-12 = C Ev G = C(v3) = C down-three
+-7-12 = C E^ G = C(^3) = C up-three
+-6-11 = C E Gv = C(v5) = C down-five
+-7-13 = C E^ G^ = C(^3,^5) = C up-three up-five
+-6-12-18 = C E G B = C7 = C seven
+-6-12-17 = C E G Bv = C(v7) = C down-seven
+-5-12-18 = C Ev G B = C7(v3) = C seven down-three
+-5-12-17 = C Ev G Bv = C.v7 = C dot down seven
+For a more complete list, see [[xenharmonic/Ups and Downs Notation#Chord%20names%20in%20other%20EDOs|Ups and Downs Notation - Chord names in other EDOs]].
 **21-tone scales:**
@@ Line 121: / Line 138: @@
 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 || Example ||= Name* ||
+||= Step Pattern ||= Cents || Example ||= Name* ||&lt; Ups/downs name ||
-||= 3, 3, 3 ||= (0)-171-343-(514) || C D E F ||= Equable diatonic ||
+||= 3, 3, 3 ||= (0)-171-343-(514) || C D E F ||= Equable diatonic ||&lt; C perfect ||
-||= 4, 3, 2 ||= (0)-229-400-(514) || C D^ E^ F ||= Soft diatonic ||
+||= 4, 3, 2 ||= (0)-229-400-(514) || C D^ E^ F ||= Soft diatonic ||&lt; C upperfect up-2 ||
-||= 4, 4, 1 ||= (0)-229-457-(514) || C D^ E^^ F ||= Intense diatonic ||
+||= 4, 4, 1 ||= (0)-229-457-(514) || C D^ E^^ F ||= Intense diatonic ||&lt; C up-2 &amp; 6, double-up-3 &amp; 7 ||
-||= 5, 3, 1 ||= (0)-286-457-(514) || C D^^ E^^ F ||= Archytas chromatic ||
+||= 5, 3, 1 ||= (0)-286-457-(514) || C D^^ E^^ F ||= Archytas chromatic ||&lt; C double-up-2, 3, 6 and 7 ||
-||= 5, 2, 2 ||= (0)-286-400-(514) || C D^^ E^ F ||= Weak chromatic ||
+||= 5, 2, 2 ||= (0)-286-400-(514) || C D^^ E^ F ||= Weak chromatic ||&lt; C double-up 2 &amp; 6, up-3 &amp; 7 ||
-||= 6, 2, 1 ||= (0)-343-457-(514) || C D^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; E^^ F ||= Strong enharmonic ||
+||= 6, 2, 1 ||= (0)-343-457-(514) || C D^&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt; E^^ F ||= Strong enharmonic ||&lt; C triple-up 2 &amp; 6, double-up 3 &amp; 7 ||
-||= 7, 1, 1 ||= (0)-400-457-(514) || C D^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; E^^ F ||= Pythagorean enharmonic ||
+||= 7, 1, 1 ||= (0)-400-457-(514) || C D^&lt;span style="font-size: 90%; vertical-align: super;"&gt;4&lt;/span&gt; E^^ F ||= Pythagorean enharmonic ||&lt; C quadruple-up 2 &amp; 6, double-up 3 &amp; 7 ||
 *these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!
@@ Line 150: / Line 167: @@
 ==13-limit Commas==
-EDO tempers out the following 13-limit commas. (Note: This assumes the val &lt; 21 33 49 59 73 78/1 |.)
+EDO tempers out the following 13-limit commas. (Note: This assumes the val &lt; 21 33 49 59 73 78 |.)
 ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||
 ||= 2187/2048 ||&lt; | -11 7 &gt; ||&gt; 113.69 ||= Apotome ||=   ||
@@ Line 233: / Line 250: @@
          &lt;td style="text-align: right;"&gt;1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;_unison_&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;unison&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;C&lt;br /&gt;
@@ Line 305: / Line 322: @@
          &lt;td style="text-align: right;"&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;perfect 2nd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;D&lt;br /&gt;
@@ Line 377: / Line 394: @@
          &lt;td style="text-align: right;"&gt;3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;perfect 3rd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;E&lt;br /&gt;
@@ Line 449: / Line 466: @@
          &lt;td style="text-align: right;"&gt;4&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;perfect 4th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;4th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;F&lt;br /&gt;
@@ Line 521: / Line 538: @@
          &lt;td style="text-align: right;"&gt;5&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;perfect 5th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;G&lt;br /&gt;
@@ Line 593: / Line 610: @@
          &lt;td style="text-align: right;"&gt;6&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;perfect 6th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;A&lt;br /&gt;
@@ Line 665: / Line 682: @@
          &lt;td style="text-align: right;"&gt;7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;perfect 7th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;B&lt;br /&gt;
@@ Line 758: / Line 775: @@
 *2: based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament&lt;br /&gt;
 *3: based on treating 21-EDO as 13-limit laconic temperament&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Chord Names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Chord Names&lt;/h1&gt;
+ &lt;br /&gt;
+Ups and downs can be used to name 21edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used.&lt;br /&gt;
+&lt;br /&gt;
+-6-12 = C E G = C = C or C perfect&lt;br /&gt;
+-5-12 = C Ev G = C(v3) = C down-three&lt;br /&gt;
+-7-12 = C E^ G = C(^3) = C up-three&lt;br /&gt;
+-6-11 = C E Gv = C(v5) = C down-five&lt;br /&gt;
+-7-13 = C E^ G^ = C(^3,^5) = C up-three up-five&lt;br /&gt;
+&lt;br /&gt;
+-6-12-18 = C E G B = C7 = C seven&lt;br /&gt;
+-6-12-17 = C E G Bv = C(v7) = C down-seven&lt;br /&gt;
+-5-12-18 = C Ev G B = C7(v3) = C seven down-three&lt;br /&gt;
+-5-12-17 = C Ev G Bv = C.v7 = C dot down seven&lt;br /&gt;
+&lt;br /&gt;
+For a more complete list, see &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Chord%20names%20in%20other%20EDOs"&gt;Ups and Downs Notation - Chord names in other EDOs&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;strong&gt;21-tone scales:&lt;/strong&gt;&lt;br /&gt;
@@ Line 764: / Line 798: @@
 &lt;a class="wiki_link" href="/augment12"&gt;augment12&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&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;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Chord Names-Triadic Harmony in 21-EDO:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&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 (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;
@@ Line 872: / Line 906: @@
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x21 equal divisions of the octave-Moment-of-Symmetry Scales in 21-EDO:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Moment-of-Symmetry Scales in 21-EDO:&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Chord Names-Moment-of-Symmetry Scales in 21-EDO:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Moment-of-Symmetry Scales in 21-EDO:&lt;/h2&gt;
   &lt;br /&gt;
 Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.&lt;br /&gt;
@@ Line 879: / Line 913: @@
 For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x21 equal divisions of the octave-Tetrachordal Scales in 21-EDO"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Tetrachordal Scales in 21-EDO&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Chord Names-Tetrachordal Scales in 21-EDO"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Tetrachordal Scales in 21-EDO&lt;/h2&gt;
   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:&lt;br /&gt;
 &lt;br /&gt;
@@ Line 893: / Line 927: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Name*&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;Ups/downs name&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 903: / Line 939: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Equable diatonic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;C perfect&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 913: / Line 951: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Soft diatonic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;C upperfect up-2&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 923: / Line 963: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Intense diatonic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;C up-2 &amp;amp; 6, double-up-3 &amp;amp; 7&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 933: / Line 975: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Archytas chromatic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;C double-up-2, 3, 6 and 7&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 943: / Line 987: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Weak chromatic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;C double-up 2 &amp;amp; 6, up-3 &amp;amp; 7&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 950: / Line 996: @@
          &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;C D^&lt;span style="font-size: 90%; 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;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;C triple-up 2 &amp;amp; 6, double-up 3 &amp;amp; 7&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 960: / Line 1,008: @@
          &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;C D^&lt;span style="font-size: 90%; 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;
+&lt;/td&gt;
+        &lt;td style="text-align: left;"&gt;C quadruple-up 2 &amp;amp; 6, double-up 3 &amp;amp; 7&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 971: / Line 1,021: @@
 The steps of these 7 basic patterns can also be permuted/rotated to give a total of 28 tetrachords, which can then be combined in either conjunct or disjunct form to yield a staggering number of scales. Thus 21edo can do reasonably-convincing imitations of the melodic forms of various tetrachordal musical traditions, such as ancient Greek, maqam, and dastgah.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x21 equal divisions of the octave-Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Rank two temperaments&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Chord Names-Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Rank two temperaments&lt;/h2&gt;
   &lt;a class="wiki_link" href="/List%20of%2021edo%20rank%20two%20temperaments%20by%20badness"&gt;List of 21edo rank two temperaments by badness&lt;/a&gt;&lt;br /&gt;
@@ Line 1,069: / Line 1,119: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x21 equal divisions of the octave-13-limit Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit Commas&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Chord Names-13-limit Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit Commas&lt;/h2&gt;
-EDO tempers out the following 13-limit commas. (Note: This assumes the val &amp;lt; 21 33 49 59 73 78/1 |.)&lt;br /&gt;
+EDO tempers out the following 13-limit commas. (Note: This assumes the val &amp;lt; 21 33 49 59 73 78 |.)&lt;br /&gt;
@@ Line 1,271: / Line 1,321: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&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;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Books / Literature:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&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:990:&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:990 --&gt;&lt;!-- ws:start:WikiTextRemoteImageRule:991:&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:991 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1008:&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:1008 --&gt;&lt;!-- ws:start:WikiTextRemoteImageRule:1009:&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:1009 --&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;
-&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Compositions/Listening:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;strong&gt;Compositions/Listening:&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Compositions/Listening:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;&lt;strong&gt;Compositions/Listening:&lt;/strong&gt;&lt;/h1&gt;
   &lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/2494" rel="nofollow" target="_blank"&gt;21-edo Trio for Organ, by Claudi Meneghin&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/2336" rel="nofollow" target="_blank"&gt;21-penny jingle, by Claudi Meneghin&lt;/a&gt;&lt;br /&gt;

21edo: Difference between revisions

Revision as of 04:20, 26 December 2016

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

Original HTML content:

Navigation menu

Search